The Mathematical Basis Of The Arts ( Joseph Schillinger, 1943)

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THE MATHEMATICAL BASIS
OF THE ARTS

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Vjuuga UNIVERSITY OF MICHIGAN

THE MATHEMATICAL BASIS
OF THE ARTS

BY

JOSEPH S^C HILLING E'R

Part One
SCIENCE AND ESTHETICS

Part Two

THEORY OF REGULARITY AND COORDINATION

Part Three

TECHNOLOGY OF ART PRODUCTION

PHILOSOPHICAL LIBRARY
NEW YORK

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Copyright, 1948, by
Frances Schillinger

All rights reserved
International Copyright Secured

PRINTED IN THE UNITED STATES OF AMERICA

to my beloved wife

Frances

for her unswerving confidence in
the permanent value of this work

J. S.

February 27, 1943

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A CKNO WLEDGMENT

The Mathematical Basis of the Arts is Schillinger's
masterwork. It represents 25 years of his discoveries
and research . This book was completed and prepared
for publication by Schillinger a year before his death
in March 1943 at the age of 47.

I wish to express my appreciation to Arnold Shaw,
executive director of the Schillinger Society, for his
editorial contribution to The Mathematical Basis of the
Arts. Mr. Shaw prepared the manuscript for the
printers and supervised the production of the book.

Mrs. Joseph Schillinger

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PREFACE

This work does not pretend to transform the reader into a proficient artist.
Its goal is to disclose th ^ me cj ranisin ofcreato rehip aa 1 it manifests itself in nature
and in the arts. This system, whicrTTfl a syiise IB itself a product of creation,
i.e. , a work of art, opens new vistas long awaiting exploration .

As the range of readers will probably vary as widely as the respective fields
covered by this theory, the author expects a correspondingly wide range of re-
actions to it. He has hopes, however, that this work will be a stimulus of high
potential, which will lead the spirit of investigation into the most majestic of all
playgrounds known .

Whereas one scientific theory overwhelms another only to be overwhelmed
by new facts and new evidence, this system overwhelms the available facts and
evidence. Hence its pragmatic validity.

Intentional art, as we know it, dates as far back as one hundred thousand
years, a comparatively brief period in hypothetically established human history.
The present system is newly born. The age of its oldest branches does not exceed
a quarter of a century which, even at the pace of present development, is a negli-
gible amount of time. Its youngest branches are only one decade old. It has
been subjected to a very limited range of experimentation and verification . Yet
evidence which has been accumulated with respect to the workability of this
system and the quality of its products is so overwhelming that, if given a chance,
it may well shatter the very foundation of the great myth of artistic creation as
we have known it since the dawn of human history.

Let this system be put to a real test. Let it penetrate into all fields of knowl-
edge, education and production for at least a generation, and then have judgment
passed upon it. The risk is negligible and the dividends too great to neglect.

February 27, 1942. Joseph Schillinger.

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CONTENTS

PART ONE: SCIENCE AND ESTHETICS

Chapter 1. ART AND NATURE 3

Chapter 2. ART AND EVOLUTION 7

Chapter 3. MIMICRY, MAGIC AND ENGINEERING 11

A. Development of Art 11

B. Evolution of Rhythmic Patterns 11

C. Biology of Sound 13

D. Varieties of Esthetic Experience @

E. History of the Arts in Five Morphological Zones ... 17

Chapter 4. THE PHYSICAL BASIS OF BEAUTY 18

Chapter 5. NATURE OF ESTHETIC SYMBOLS 23

A. Semantics 23

B. Semantics of Melody 25

C. Intentional Biomechanical Processes 28

D. Definition of Melody 29

Chapter 6. CREATION AND CRITERIA OF ART 30

A. Engineering vs. Spontaneous Creation 30

B. Nature of Organic Art 32

C. Creation vs. Imitation 34

Chapter 7. MATHEMATICS AND ART (5|

A. Uniformity and Primary Selective Systems (SjL

B. Harmonic Relations and Harmonic Coordination . . . (4u

C. Other Techniques of Variation and Composition . . . (4$

D. Pragmatic Validity of Theory of Regularity @)

PART TWO: THEORY OF REGULARITY AND COORDINATION

Chapter 1. CONTINUUM 51

A. Definition of an Art Product by the Method of Series . . (5})

B. Parametral Interpretation of a System 56

C. The First Group of Art Forms 58

D. Time (X«) as a General Parameter 71

E. The Second Group of Art Forms 74

F. The Third Group of Art Forms 78

G. Correspondences Between Art Forms ' 80

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CONTENTS

Chapter 2. CONTINUITY 85

A. Series of Values 85

B. Factorial-Fractional Continuity 92

C. Elements of Factorial-Fractional Continuity 94

D. Determinants 98

Chapter^) PERIODICITY 109

A. Simultaneous Monomial Periodicities 112

B. Polynomial Relations of Monomial Periodicities . . . 123
(O. Polynomial Relations of Polynomial Periodic Series . . 129

MJ. Practical Application in Art 140

JL Synchronization of the Second Order 141

Periodicity of Expansion and Contraction 146

G. Progressive Symmetry 157

Chapters PERMUTATION . . . 158

A. Displacement 158

B. General Permutation 162

C. Permutations of the Higher Orders 168

D. Mechanical Scheme for the Permutation of Four Elements 172

Chapter 5. DISTRIBUTIVE INVOLUTION 173

A. Powers 173

B. General Treatment of Powers 179

Chapter 6. BALANCE, UNSTABLE EQUILIBRIUM, AND

CRYSTALLIZATION OF EVENT 184

A. Formulae 186

B. Unstable Equilibrium in Factorial Composition of Duration

Groups 189

Chapter 7. RATIO AND RATIONALIZATION 193

A. Ratio — Rational — Relation — Relational 193

B. Rationalization of the Second Order 193

C. Rationalization of a Rectangle 195

D. Ratios of the Rational Continuum 214

Chapter 8. POSITIONAL ROTATION 215

A. Dimensionality of Positional Rotation 215

Chapter 9. SYMMETRY 219

A. Symmetric Parallelisms 219

Esthetic Evaluation on the Basis of Symmetry .... 223
C. Rectangular Symmetry of Extensions in Serial Develop-
ment 224

Chapter 10. QUADRANT ROTATION 232

Chapter^. COORDINATE EXPANSION 244

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CONTENTS ix

Chapter 12. COMPOSITION OF DENSITY 254

A. Technical Premise 255

B. Composition of Density-Groups 256

C. Permutation of the Sequent Density-Groups within the

Compound Sequent Density-Group 260

/D) Phasic Rotation of A and A - * through t and d . . . . 262

Practical Application of A - * to S - * 266

PART THREE: TECHNOLOGY OF ART PRODUCTION

Chapter 1. SELECTIVE SYSTEMS 273

A. Primary and Secondary Selective Systems 273

/B, Temporal Scales 274

C. Pitch Scales 277

D. Scales of Linear Configurations and Areas 284

( 1 ) Periodicity of Dimensions 284

(2) Periodicity of Angles 293

(3) Rectilinear Segments Forming Angle's in Alternating

Directions . . . 302

(4) Monomial, Binomial and Trinomial Periodicity of

Sector Radii 313

(5) Periodicity of Radii and Angle Values 314

(6) Periodicity of Arcs and Radii 324

(7) r 4 ^ 3 Rhythmic Groups in Linear Design .... 329

(8) Planes 336

(9) Closed Polygons 344

E. Color Scales 346

Chapter 2. PRODUCTION OF DESIGN 363

A. Elements of Linear Design 363

B. Rhythmic Design 365

C. Design Analysis 375

D. Three Compositions in Linear Design 391

E. Problems in Design 397

Chapter; 3. PRODUCTION OF MUSIC 399

— f A\ Coordination of Temporal Structures 399

B. Distribution of a Duration Group 401

C. Synchronization of an Attack Group 402

D. Distribution of a Synchronized Duration Group .... 404

E. Synchronization of an Instrumental Group 406

Chapter 4. PRODUCTION OF KINETIC DESIGN 414

A. Proportionate Distribution within Rectangular Areas . . 414

B. Distributive Involution in Linear Design 418

C. Application to Dimensions and Angles 423

D. Positional Rotation Applied to Kinetic Design .... 423

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Chapter 5. PRODUCTION OF COMBINED ARTS 429

A. The Time- Space Unit in Cinematic Design 429

B. Correlation of Visual and Auditory Forms 432

APPENDICES

Appendix A. BASIC FORMS OF REGULARITY AND COORDINA-
TION 447

I. Binary and ternary synchronization of genetic factors . . 447

(A) Binary synchronization 447

(B) Binary synchronization with fractioning .... 458

(C) Ternary synchronization 476

(D) Generalization : synchronization of n generators . . 476

II. Distributive involution groups 502

A. Distributive square of binomials 502

B. Distributive square of trinomials 520

C. Generalization : distributive square of polynomials . . 542

D. Distributive cube of binomials 563

E. Distributive cube of trinomials 568

F. Generalization : distributive cube of polynomials . . 586

III. Groups of variable velocity 636

Appendix B. RELATIVE DIMENSIONS 643

Appendix C. NEW ART FORMS 663

I. Double Equal Temperament 665

II. Rhythmicon 665

III. Solidrama 667

Appendix D. POETRY AND PROSE 669

Appendix E. PROJECTS 671

I. Books 672

II. Instruments 673

Glossary (Compiled by Arnold Shaw) 675

Index 687

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PART ONE
SCIENCE AND ESTHETICS

This Theory is Based on the Following Postulates:

J. The fertility of a postulate.

2. Uniformity as the basic concept.

j. Fractioning of unity as the potential of evolution.

4. Unstable equilibrium as a genetic force.

5. The principle of interference as a factor of growth and evolution.

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CHAPTER 1

ART AND NATURE

TFart implies selectivity, skill and organization, ascertainable principles must
underly it. Once such principles are discovered and formulated, works of art
may be produced by scientific synthesis. There is a common misunderstanding
about the freedom of an artist as it relates to self-expression. No artist is really
free. He is subjected to the influences of his immediate surroundings in the
manner of execution , and confined to the material media at his hand . If an artist
were truly free, he would speak his own individual language. In reality, he
speaks only the language of his immediate geographical and historical boundaries.
There is no artist known who, being born in Paris, can express himself spontane-
ously in the medium of Chinese 4th century a.d., nor is there any composer,
born and reared in Vienna, who possesses an inborn mastery of the Javanese
gamelan. 1

The ke> to real freedom and emancipation from local dependence is through
scientific method. Authors, painters and composers have exercised their imagi-
nations from time immemorial. And yet can any of their most daring dreams
compare with what science offers us today? Man has always flown in his dreams;
nevertheless, these never satisfied his urge for "real" flight. Since antiquity, a
number of myths has persisted of man's attempts to fly by means of artificial
wings. Such flights have always failed. Let this be a lesson to artists. We
cannot liberate ourselves by imitating a bird. The real way to freedom lies in
the discovery and mastery of. the principles of flight. Creation directly from
principles, and not through the imitation of appearances, is the real way to free-
dom for an artist. Originality is the product of knowledge, not guesswork.
Scientific method in the arts provides an inconceivable number of ideas, technical
ease, perfection, and, ultimately, a feeling of real freedom, satisfaction and
accomplishment.

My life-long study, research, and accomplishments as a creative artist have
been devoted to a search for facts pertaining to the arts. As a result of this work
I have succeeded in evolving a scientific theory of the arts. The entire system
emphasizes three main branches 2 :

1 . The Semantics of Esthetic Expression

2. The Theory of Regularity and Coordination

3 . The Technology of Art Production

'The gamelan (or gamelang) is a primitive type in Part I, particularly Chapter 5. The reader is

of orchestra, native to Java, and consisting of an also referred to The Schillinger System of Musical

instrument which resembles our xylophone, and Composition, which offers basic ideas in Book XI.

several small buffalo hide drums, which are strum- pp. 1410-1477. Although the last-mentioned

med rather than struck with the ringers. (Ed.) pages deal with the "Semantic Basis of Music,"

*The Theory of Regularity and Coordination is they present fundamental aspects of the relation-
Part II of the present work. The Technology of ship between expression and geometrical forms.
Art Production is Part III. Materials relating to emotional patterns and spatial configurations.
The Semantics of Esthetic Expression will be found (Ed.)

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SCIENCE AND ESTHETICS

The Semantics of Esthetic Expression deals with the relationship between
form and sensation, and with the associational potential of form, thus establish-
ing the meaning of esthetic perception . The Theory of Regularity and Coordina-
tion discloses the basic principles of creatorship. The Technology of Art Production
embraces all details pertaining to the analysis and synthesis of works of art in
individual and combined media. The second branch is more voluminous than
the other two combined, and requires years of study.

This theory of art is not limited to the conventional forms but embraces all
the possible forms that can be evolved in space and time, and perceived through
the organs of sensation. In addition to individual art-forms, it discloses all the
possible compound art-forms, and all forms of technique that make possible
the transformation of individual art-forms into compound ones.

According to this theory, art is determined as a logical system in the Carte-
sian and Einsteinian manner 8 , i.e., as a system of correlated parameters (measur-
ing lines) . A work of art may be adequately expressed by means of graphs afford-
ing both analysis and synthesis. Each individual graph may express a special
art component (such as pitch in sound, hue in color, etc.) in relation to time.

The laws of rhythm, formulated in this theory as general esthetic laws, are
based on two fundamental processes:

(1) the generation of harmonic groups through interference 4 :

(2) the variation of harmonic groups through combinatory and involution-
ary techniques.

Certain conclusions drawn concerning the esthetic properties of art phe-
nomena coincide with discoveries made concerning the harmonic structure of
crystals (Goldschmidt, Whitlock) and the properties ' of tangent trajectories
(Kasner). This can only mean that:

(1) either there are general laws of the empirical universe in which esthetic
realities take their place among the physical; or

(2) our method of mathematical deduction, being limited by the laws of
its own logic, cannot be divorced from the object analyzed.

In either case, we are bound to cancel the line of demarcation between
esthetic and physical realities. The history of art may thereupon be described
in the following form:

(1) Nature produces physical phenomena, which reveal an esthetic har-
mony to us; this harmony is the result of periodic and combinatory
processes; esthetic actualities embody mathematical logic. This is the
pre-esthetic , natural (physical, chemical, biological) period of art crea-
tion.

"The distinction implied here relates to the num-
ber of coordinates required to represent the differ-
ent art forme. In Chapter 1. Continuum, of Part
II, Schillinger classifies the individual art forms
into eighteen groups depending on the sense or-
gans they affect. Each art requires a system
of special components (light, sound, mass, or sur-
face, etc.) to represent it and from one to four
general components (time = X4, space = xi, xj,
x«). The Cartesian system, as distinguished

from the Einsteinian, involves only two coordi-
nates. (Ed.)

*The concept of interference is pivotal in Schll-
linger'8 approach to the arts. Arising from phe-
nomena observed in all fields of wave motion, it
refers to the coincidence of two waves which
results in a new summary wave. This concept
(and a graphic technique based on it) is developed
in detail in Chapter 3. Periodicity, of Part II,
and exemplified in Appendix A. (Ed.)

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(2) Man recreates esthetic realities by reproducing the appearance of the
physical realities through his own body, or through a material at his
command; this process of reproduction involves mathematical logic,
regardless of whether the artist is conscious of it or not; imitating nature
in a material medium, he expresses the laws of mathematical logic
through his sensory experience; this is the intuitive period of art creation.

(3) Becoming more and more conscious in the course of his evolution, man
begins to create directly from principles; with developments in the
technique of handling material art media (special components) and the
rhythm of the composition as a whole (general components: time,
space), man is enabled to choose the desired product and allow the
machine to do the rest; this is the rational and functional period of art
creation .

Thus, the evolution of art falls into a closed system. An esthetic reality may be
either a natural product, a product of human creative intuition, or a product of
scientific synthesis, realized through computation by mathematical logic. In
actuality, all three aspects coexist in perpetual interaction.

Every work of art conceived and executed by man is a modified (often merely
reflected) counterpart of actuality. Music, for example, is a man-made illusion
of actuality, and so is every art. Music is merely a mechanism simulating organic
existence. Music makes one believe it is alive because it moves and acts like
living matter. Even Aristotle had observed that "rhythms and melodies are
movements as much as they are actions." The common belief that "music is
emotional" has to be repudiated as a primeval animism, which still survives
in the confused psyche of our contemporaries. This erroneous conception can
be easily justified as "naive realism." Music appears emotional merely because
it moves — since everything that moves associates itself with life and living.
Actually, music is no more emotional than an automobile, locomotive or an air-
plane, which also move. Music is no more emotional than the Disney characters
that make us laugh, but whose actual form of existence is not organic, but me-
chanical (a strip of pictures drawn on celluloid and projected on a screen).

Everything that moves is a mechanism, and the science of motion is me-
chanics. The art of making music consists in arranging the motion of sounds
(pitch, volume, quality) in such a manner that they appear to be organic, alive.
The science of making music thus becomes the mechanics of musical sounds.
The technique of this science enables the art of music to serve its ultimate purpose:
the conveyance of musical ideas to the listener .

- Nature is the source and supplies the media and the instruments of the arts.
The sources of the art of painting are the forms, the texture, and the coloring of
rainbows, sunsets, birds' plumages, crystals, shells, plants, animal and human
bodies. Minerals, plants and vegetables are the media (pigments) while the sense
of vision is the instrument of that art.

Thunder, animal sounds, and the echo are as much the sources of music
as all the inorganic and organic forms that provide the structural patterns for
musical intonation and continuity. Lungs and vocal cords, reeds and animal

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SCIENCE AND sESTHETICS

skins, as well as electricity, are the media (sound-producing devices) of the art
of music; and the sense of hearing is the instrument of the art.

Natural forms originated as a necessity: as instruments for efficient exist-
ence. Multiplication of forms and images, through optical and acoustical reflex-
ion, and circumstantial mimicry, which provides aggressive and protective size,
offensive and defensive shape and coloring, constitute the first steps in the evolu-
tion of art forms. Deduction of esthetic norms, combined with imitation and
readjustment of appearances according to these norms, constitutes the succeeding
step — intentional mimicry. The final step in the evolution of the arts is the
scientific method of art production, whereby works of art are manufactured and
distributed according to definite specifications. This final step becomes possible
only after the laws of art have been disclosed.

Discovery of the laws of art has been an old dream of humanity. In the
Li-Ki, or Memorial Rites of the Ancient Chinese, we read: "Music is intimately
connected with the essential relations of beings. Thus, to know sounds, but not
airs, is peculiar to birds and brute beasts; to know airs, but not music, is peculiar
to the common herd; to the wise alone it is reserved to understand music. That
is why sounds are studied to know airs, airs in order to know music, and music
to know how to rule."

The science of art-making must be concerned with two fundamentals:

(1) the mechanics of pattern-making 5 ; and

(2) the mechanics of reactions. 6

A scientific theory of the arts must deal with the relationship that develops be-
tween works of art as they exist in their physical forms and emotional responses
as they exist in their psycho-physiological form, i.e., between the forms of excitors
and the forms of reaction. As long as an art-form manifests itself through a phys-
ical medium, and is perceived through an organ of sensation, memory and
associative orientation, it is a measurable quantity. Measurable quantities are
subject to the laws of mathematics. Thus, analysis of esthetic form requires
mathematical techniques, and the synthesis of forms (the realization of forms
in an art medium) requires the technique of engineering.

There is no reason why music or painting or poetry cannot be designed and
executed just as engines or bridges are. Today's technical progress offers ample
evidence of the achievement possible through the method of engineering, i.e.,
through the method of expedient economy and efficiency. And if this method
has transformed the most daring dreams of yesterday into the actualities of today,
it can be equally successful in the field of art.

'The mechanics of pattern-making are described tionship between the form9 of excitors and the
in Parts II and III of the present work. (Ed.) forms of emotional response, or the semantics of
•The mechanics of reactions refers to the rela- expression. See footnote 2 above. (Ed.)

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CHAPTER 2

ART AND EVOL U TION

"V\ TK usually tKink of art in terms of local dependence. Such an approach
produces many "connoisseurs" and "experts." There is always some-
body who knows "everything" about Sappho, Duerer, Shakespeare, Cezanne, or
Surrealism. But as folk-wisdom suggests: "One cannot see the woods for the
trees." What we want to know is art's fatal history, its morphology .

If we possessed some reliable information concerning the mechanics of fate,
it might offer a definite clue to the solution of our problem. But as we have no
such knowledge, we must be satisfied with the only reliable kind of information
we can gather in any field — that achieved through application of scientific method .
But scientific method furnishes only one kind of information on any object: the
knowledge gathered from its behavior. What the object is ultimately, what its
meaning is — we do not profess to know. But we may observe its actions and
draw logical conclusions after such observation.

If we could trace the original parent-forms of what we know as the arts
today, we might discover, through observation of these forms, evolutionary
tendencies as they manifest themselves in the behavior of such forms. Then by
further application of logic, we might reconstruct from such tendencies the
morphological image of the whole. Art history is too young to let us see the goal;
nevertheless, the evidence accumulated so far is sufficient to reveal some of its
tendencies.

If we had developed a science of events, a sort of fatal technology or "event-
ology ," then the destiny of the arts might be known to us. But today's theoretical
sciences seem to be losing ground; a "theory of uncertainty" usurps the place of
the theory of probability.

The pragmatic sciences, however, continue to function satisfactorily in the
fields in which they have been evolved. For example, knowledge of topography
permits us to locate a certain geographical point with any degree of precision re-
quired for practical purposes. But to locate an event in an evolutionary chain,
which is either finite or infinite, is quite a different matter. The subject of our
investigation is the morphology of creatorship, which should include all forms of
inventiveness, i.e., both the scientific and the artistic. This implies the necessity
of a "general theory of regularity." But up to now we have not had such a
theory; thus, we are unable to formulate the kind of regularity which controls the
appearance of new ideas in our world.

Professor Dirac, a British mathematician, suggested during the summer of
1937, that there are new numbers constantly appearing in the universe. We
know that the classical civilization of Europe did not possess the concept of a
million or billion. Even within the 10,000 limit, their expressions were extremely
cumbersome. Our laws of esthetics must necessarily be cumbersome for the
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present, as we have never had any general theory explaining esthetic theories as
an evolutionary group. Yet we are able to deduce a scientific theory of art-making
from the manifold of the art experiences of past cultures.

The span of the human race is an insignificant episode in world history.
The span of the arts is only an insignificant episode in the history of the human
race. The Age of Mammals, grass and land forests is about one-tenth of the en-
tire period of life on our planet, starting with sea scorpions. Out of twenty mil-
lion years of the Cainozoic age, the human race starting with the pithecanthropus
erectus covers only about five hundred thousand years — that is, about one-for-
tieth of the whole period of existence of vertebrate mammals. According to re-
cent discoveries (1936), some signs of intentional design have been discovered
in the Himalayan region. These are supposed to go as far back as one hundred
thousand years ago. In other words, the earliest art traced so far spans not more
than one-fifth of the history of the human race. European art, if it is to be reck-
oned from the designs found in the caves of the early Heidelberg dwellers (Nean-
derthal Man), is about twenty-five thousand years old, or about one-fortieth of
the history of the human race.

The human race is confined to five senses and associative orientation. Art
forms are perceived through the five senses and stimulate associative impulses.
The senses impose limitations on art materials. The materials of the tonal art,
for example, are limited to low frequencies and low amplitudes. Amplitudes may
be magnified but the possible range of frequencies depends entirely on further
evolution of the sense of hearing. While the most developed organ of sensation
in the human race today is sight, the sensation of hearing is very limited as to
intensity and the range of frequencies. We do not hear anything beyond 18,000
vibrations per second. In dealing with higher frequencies, sound decreases in
volume; instead of increasing in pitch, it fades out, remaining on a certain high
pitch. Without electrical amplifiers, sound cannot be transmitted for any dis-
tance comparable with the distance covered by our vision. Ordinary human
speech cannot be heard even a mile away. The loudest symphony orchestra
playing outdoors and surrounded by silent areas for miles cannot be heard even
for a distance of five miles. Sound wave frequencies, from sixteen to about five
thousand, compose only an infinitesimal group within the range of all types of
wave motion — those producing colors, heat waves and various forms of radiation.

Art appears at first as a necessity. Ornamental tendencies develop much
later. Art will cease to exist when there is no need for it. The latest anthropo-
logical conclusions are that the mind develops to a much greater extent than the
organs of sensation to serve the ends of progress and evolution. Animals have
keener organs of sensation than ours because these are their tools for protection
and defense. A human being may be warned by a telegram about a coming dan-
ger. An animal has to rely entirely on its senses.

Though in our empirical existence we deal with a world crowded with matter,
the emptiness of the universe is beyond our imagination. Likewise, the quantity
of "tonal matter" is very limited in the sensory continuum. Forms of energy
transformed into matter are comparatively rare. Space saturated with matter
is t analogous to time saturated with events. We live in a world crowded with

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ART AND EVOLUTION

9

events. The space-time continuum of music consists of the alternation of silence
and sounds. Soundless time empirically seems to be longer. Time saturated with
sound variations produces the opposite effect. Concentration of musical matter
in the time-space continuum is structural energy in sound. This means that the
flow of time as perceived in a musical composition speeds up or slows down in
direct relation to its structural constitution. A short musical composition con-
taining many recurrences becomes unbearable. Our esthetic experience tells us
that a short composition must sound longer in order to seem satisfactory and
complete. A long composition must produce an illusion of being considerably
shorter than its actual duration . These time sensations are common in our every-
day experiences. The lack of activity and events about us makes time move at a
slower pace than when we encounter a multitude of events. Moments of satis-
faction , pleasure and ecstasy always seem too short. Moments of boredom always
seem too long. Crowded events which assume simple periodic forms and do not
require active participation on our part, lead us to boredom, that is, to time
slowing down.

The mental growth of humanity, as revealed in scientific thinking, may be
stated as a tendency to unite seemingly different categories into a complex unity
into which previous concepts enter as component parts. The evolution of
thought is a process of synthesizing concepts. Creation from principles should
not be confused with imitation of appearances (mimicry).

It is time to admit that esthetic theories have failed in the analysis as well as
synthesis of art. These have been unsuccessful both in interpreting the nature
of art and in evolving a reliable method of composition. The artistic approach to
art has proved to be inadequate in solving the problem of creative experience.
An adequate method has to be found. But methodology is a scientific, not an
artistic development. The evolution of method is the ultimate goal of science.
Thus, science comes to the rescue of art. "Esthetic qualities" can be detected
within art material, transformed into the geometrical relations of its components,
" and finally into corresponding number values. The entire problem of art empha-
sizes three phases:

(1) Scientific (graph) method of recording a work of art 1 through its compo-
nents (in place of the present, inadequate systems of artistic notation),
which permits us to measure, to analyze, to draw conclusions, and to
deduce norms;

(2) Modification of a work of art through variation of its inherent geo-
metrical properties in a corresponding graph record;

(3) Production of a synthetic work of art from a system of number values,
transformed into geometrical relations, and, finally, into corresponding
components of art material.

Thus, science establishes a precedent for the development of an art theory as a
system satisfactory in any special case. Scientific method penetrates into another

'In music, for example, the horizontal axis rep- axis represents pitch, with each square equivalent
resents note durations (each square may equal an to a semitone or a tone as the case may be. (Ed.)
eighth note, a sixteenth note, etc.) The vertical

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SCIENCE AND ESTHETICS

realm of the unknown, and establishes premises for the analysis and synthesis
of art as creative experience.

Scientific laws, such as the law of gravity, make prediction possible. Art,
being an evolutionary group, must function through the laws of evolutionary-
groups. The differentiation of art forms corresponds to differentiation of th?
senses. Structural and associative pattern-making is universal. Art forms con-
sist of structural as well as associative patterns. All evolutionary groups reveal the
tendency of acceleration. The evolution of the human race as well as of our
planet presents such evidence, and art is no exception to this law.

A structural evolutionary group may be expressed in the following concept
series: impetus — motion — inertia — balance — stabilization — crystallization
— deposition — disintegration (transformation). A pentacle in a starfish is a
pattern crystallized for efficient existence. An abstract pentacle (geometrical
pattern) becomes a source of new functional association, that is, it becomes the
symbol of a fighting unit (the Red Army). It involves geological and biological
as well as esthetic patterns. The appearance of new biological and esthetic pat-
terns is necessitated by readjustment. Pattern-making has its general source in
electro-chemical patterns of brain functioning.

According to Professor Barr, Yale anatomist, "Physiology becomes a branch
of electrical engineering" (1936). Thus, the geometry of thought becomes the
source of universal pattern-making. This bio-geometrical generator asserts
certain tendencies, which in turn produce certain configurations and certain colors.
Perhaps in the near future, we may learn that creative experiences are merely
geometrical projections of the electro-chemical patterns of thought on various
materials having sensory effects upon us.

The mysterious character of the prime number has always been a source of
fascination to primitive cultures. This is probably why such numbers have been
emphasized in the case of the Hindu (17), Javanese (5), and Siamese (7) musical
cultures. Our musical culture deals with the number 12 — and all kinds of irrele-
vant reasons and excuses have been found for this choice. The real justification
for the value of 12 is its versatility with respect to division. It is the smallest
number up to 60 which contains so many divisors. Versatility in division and
other quantitative properties lead to greater combinatory variability. Theories
a posteriori are very characteristic forms of art theories in general. Offering
nothing in the analysis of the creative processes of art, such theories expose theii
futility in the contention that a genius is above theories, and that his creativity
is free and does not conform to any laws or principles. This is the mythological
period of esthetics. In 2000 B.C. lightning was a revelation of divine power and
could not be explained in terms of human experience. Yet it was produced every
hour on schedule at the New York World Fair of 1939, and was offered as a form
of entertainment. There is less and less room for mystery and divinity so far
as the manipulation of material elements is concerned .

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CHAPTER 3

MIMICRY, MAGIC AND ENGINEERING 1

A. Development of Art

TNTEGRATION of esthetic experience assumes the following evolutionary
cycle: mimicry (passive transformation), magic (active transformation), and
engineering (scientific transformation). We shall illustrate this proposition in
its application to a concrete art form.

The transformation of matter into energy and the transformation of a sensa-
tion into a concept find their analogues in the history of music. The development
of musical instruments and the performance of music, as well as the development
of the forms of musical composition and theories, follow a similar process of de-
materialization; from the first intentional sound-production by means of the
bodily organs, through the most elaborate material instruments (piano, organ),
finally, to dematerialized electronic instruments. From reliance upon the organs
(lungs, vocal cords, diaphragm, lips, fingers, arms, etc.) as the agents of perform-
ance, through utilization of electrical devices for the development of volume
and tone-quality, finally, to elimination of the performer. From unintentional
improvisation and imitation, through highly developed-artistic creation, finally to
scientific creation and engineering with automatic production of music and elimi-
nation of the composer. From spontaneous forms induced by biological pattern-
making, through scholastic theories of rules and exceptions, finally to a scientific
theory dealing with laws of intentional creation, and developed in accordance
with general science: that is, from biological to mathematically logical patterns.

There are three fundamental periods in the history of musical instruments.
First: a mammal or a man uses the organs of his body, vocal cords, palms, wings,
etc. Second: man begins to utilize the objects of the surrounding world: cockle-
shells, horns, bamboo stems, etc. Through imitation of the appearances and
materials, he tries to reproduce similar instruments. It takes centuries, if not
millennia, to improve these crude forms offered by nature. Third: man discovers
scientific methods of sound production. At the early stage of this period, he tries
to utilize his physical knowledge of sound in order to improve existing instruments.
At a later period he learns to produce sounds directly from oscillatory sources,
such as electro-magnetic induction, interference in the electro-magnetic field, etc.
At the same time he tries to increase the area covered by sound production. He
develops ways and means of amplifying sound and of broadcasting.

B. Evolution of Rhythmic Patterns

All rhythmic patterns evolved by the human race, from prehistoric time
until now, do not exceed the 12/12 series: that is, twelve is the largest genetic

'The material contained in this chapter was printed by permission from the M. T. N. A.
presented in part before meetings of the Music Volume of Proceedings for 1937. (Ed.)
Teachers National Association in 1937. It is re-

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SCIENCE AND ESTHETICS

factor necessary to produce all the existing and known rhythmic patterns. Syn-
chronized simple periodic groups of different frequencies generate the original
unbalanced binomial, whose major genetic factor (generator), being the sum of
the binomial, becomes the determinant of the series. The interference of the
original binomial against its own converse generates a new resultant (which is a
trinomial) 1 .

Permutation of members of this group produces new derivative groups.
These derivative groups fall into synchronization with their own permutations
and produce the next interference group. The process follows in the same order
until one of the interference groups produces uniformity. From this point on,
the entire set of processes repeats all over again in the form of distributive
squares. Any further distributive involution group may be assumed as a limit.
The ultimate limit-group produces a hypothetical absolute uniformity. These
compound processes migrate from one series to another, often not developing
beyond the second power. The law of involution works in both coordinates, thus
producing simultaneous and sequent groups.

Transition from one series of patterns to another takes as long a period as
the growth and decline of a civilization. The evolution of chord structures took a
very short period as compared to the evolution from unison to chord structures.
It took about one hundred and fifty years to adopt a seventh-chord (tetrad)
after the fifth-chord (triad) was known for centuries. It took about fifty years
to adopt a ninth-chord (pentad) after the seventh-chord was assimilated. In
the last few decades, the art of music has evolved more varied chord structures
than in the entire past history of the human race.

While in the classical period of European history, the evolution of new mus-
ical systems did not occur more frequently than at intervals of a century, today
so many systems are being evolved every decade that even the slightest attempt
to apply them to the practice of musical composition would take a longer period
than the entire period of the history of the human race. There is a decided rate
of acceleration in all evolutionary processes. This applies to biological types
as well as esthetic types. Perhaps it took more than a million years from the
first sigh of the first mammal until the first intentional melody was devised. It
took perhaps a few millennia until polyphony was devised, while in our day there
are dozens of symphonies begun daily, each one trying to develop new experi-
mental or theoretical approaches.

When existing tuning systems seem to lose their freshness, composers search
for new intonations. Debussy was attracted by the Melanesian intonations
while Ravel tried to adapt Madagascar songs. The alert system-makers of today
offer quarter-tone and other devices. With all this immense raw material, a
system of esthetic utilization becomes more and more urgent. It is more essential
for the sake of esthetic efficiency to be able to manipulate in new ways two or
three familiar elements than to get lost in a rich and variegated group of new
elements that have not passed through our previous experience.

'The successive processes referred to in this
paragraph and in the succeeding paragraphs are
described in Part II, Chapter 3, Periodicity. (Ed.)

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C. Biology of Sound

We have already referred to motion as the source of pattern-making. Mus-
cular tension and release is one of the first sources of organic sound. Though
commonly unknown and generally repudiated when brought into a discussion,
this fundamental aspect of musical semantics had already been known to Aris-
totle who referred to rhythms and melodious sequences as movements. This is the
first penetration into the true nature of musical language. Animal sound con-
tains all the components of tonal art: intensity, frequency and duration.

The biological factors of sound are:

1 . Reaction of an organism to sound as a signal of movement.

2. Connection between increase and decrease in intensity of sound, and

Movement itself is the first source of music: periodic vibrations occurring in
nature produce sound — the material of music; organic movements (breathing,
locomotion, expansion, contraction) produce the forms of music. The mechan-
ical constitution of music varies with times and places, yet the patterns of it are
familiar to us from our bio-mechanical experiences.

When we arrive at a conception of pattern-making as an experience general
to all the perceptible world, musical phenomenon becomes merely a special
case of esthetic phenomena in general. Its distinction from other esthetic phe-
nomena depends not so much on the actual patterns as on the material in which
these patterns are realized. Musical patterns do not necessarily signify the art
of music. They may be created by a group of circumstances and not by the
intentions of an individual or a collective artist. Thus, musical form may result
from personal as well as impersonal expression.

The natural sources of music are in sounds as well as in the patterns of the
organic and inorganic worlds. In the early history of mammals, sound probably
was a spontaneous reflex of vocal cords, induced by fear and stimulated by the
contraction pattern as a geometrical expression of fear. This sound became a
signal of approaching danger. The process of crystallization itself was the result
of repeated experiences through which the mammal learned of its efficiency.
Evolution of the art of music from a signal has been substantiated by Karl Stumpf
in The Origin of Music. A sound signal coordinates group reactions.

Here we have the origin of the organizing power of music. Efficiency (order,
organization) results from two opposite processes: aggression (attack) and fear
(defense). Thus, we acquire all the organizing forms of music: hunt, regimenta-
tion, emergency and labor signals. Hence, the deification of music as an organ-
izing power. Music becomes a magical factor. By means of a sound signal,
an animal tries to induce fear in another animal. This is the first source of the
incantation of evil. If a sound signal can counteract the unfavorable and evil, it
probably can attract the favorable and the good . Evocation of the favorable is
the first religious function of music. After a time, primitive incantations are dis-
sociated from their original magical connotations and disintegrate at the end
of their evolutionary cycle into operatic, pseudo-mystical and nursery-rhyme
forms.

analogous variations of intensity in the organism.

Digit

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If music has an influence upon the evil in the surrounding world, it may have
the power to influence this evil in human bodies. Hence, the medicinal applica-
tion of music through the course of many centuries. Music as a healing device
penetrated, not only into such fields as psychotherapy, but into gynecology as
well. Today forms of treatment by means of sound waves are being extended.
Scientifically speaking, the difference between treatment by means of low fre-
quency waves (sound waves) is only quantitatively different from treatment by
means of waves of high frequency (x-rays).

D. Varieties of Esthetic Experience

Music as an idea-forming factor has been known since Plato, Aristotle and
Aristoxenes. Plato in his Politeia discusses music as an ethical factor, and asserts
that the purely emotional enjoyment of music is inherent in slaves as well as in
animals. It was a part of the school curriculum at that time to know which
musical scales stimulated virtues. Some of the scales were rejected because
they had a bad influence upon the young generation. We have not progressed
much since then. We meet people in our own society today who believe that
certain patterns in musical scales have bad influences on our generation. They
have in mind certain hybrids between the ecclesiastic and religious music of
England, and the music of African cannibals. Apparently, this ethically injuri-
ous music is so alluring that it affects not only the "drifting" young generation,
but some of the greatest composers of our time as well .

Contemplative music has its origin in the disintegration of labor processes.
It is a form of movement by inertia or minute stimuli. Such are pastorals, barca-
rolles, and cradle songs. This is the music of satisfaction and of contemplation,
that is, the lyrical form of ordinance. What is an obsession, caused by fear of
unknown mysterious forces in a primitive man, assumes the form of obsession
by forces that still contain a certain amount of mystery for the civilized man.
Love is one of such forces. The active and passive forms of this obsession are
nocturnes, love songs and serenades. Forms of dissatisfaction and unbalanced
existence stimulate readjustment. Readjustment calls for organization and
sometimes revolt. The expressions of dissatisfaction and revolt are revolutionary
hymns and songs.

The evolution of ecclesiastic music into pure music assumed the following
pattern: crystallized ecclesiastic dogmas influenced music patterns directly and
indirectly, thus becoming esthetic dogmas. The admiration of divine harmony
as a form of perfection resulted in admiration of musical harmony that would
sound perfect to the human ear. Thus, the cult of concord was created. The
evil of the primitive man assumed the form of dissonant chords for the civilized
man. Music began to seek formal purity and became art for art's sake. From
the bewitching concept through the glamorous, beautiful, charming, pretty, ele-
gant, gallant, neat and orderly stages went the disintegration of musical patterns.
Form became a crystallized scheme. Deposition and disintegration are the out-
come of this evolutionary group. The cult of craftsmanship transforms into form-
alism and scholasticism , and leads to a dead end of musical theory and practice.

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IS

There has always been extensive speculation concerning the nature of music
structures. Pythagoras attributed the meaning of music to the motion of celes-
tial bodies. In the eighteenth century Saint-Martin compared the tones of a
major triad with doubled root to the four elements. Schopenhauer, Novalis,
Spencer and others tried to link music with architecture, poetry, and the processes
of life itself.

There are many views on what music is supposed to represent. From its
original medicinal connotations, music deviates into various influences in the
field of psychology. Music often serves as a release of psychological obsessions.
In other cases, music itself becomes an obsession. Frequently, musical abilities
develop on account of other abilities. There are many musicians with subnormal
mentality as well as people who are insane in the medical sense of this word but
who possess extraordinary musical abilities and almost supernaturally retentive
musical memories. In relation both to instruments and esthetic forms, musical
trends are dependent upon sociological, economic and technical forms. These
often determine the velocity make-up of the music of a corresponding era.

The e ducatio naj_yalue_of music lies in the field of_technje_al routines. In
learning to play an instrument, an individual acquires the agility and the coordi-
nation of his muscles and respiratory technique. By writing and analyzing music,
and studying intelligent music theories, an individual acquires similar agility and
coordination of his mind. Rational musical education is more important than
the immediate acquisition of one type of routine, which may be useless ten years
later. The education of a professional musician must include all the technical
training possible, combined with a thorough knowledge of sound as material,
and a complete understanding of the general methods involved in all musical
procedures. Musical instruments as well as musical forms go through their con-
tinuous evolution. It may happen that in the future neither finger agility nor
sound-production will be necessary any longer.

It is the varieties of creative experience in music that makes the art of musical
composition so intangible. Music may be composed in a rational as well as irra-
tional way. The extreme of the latter is music appearing in a dream where the
element of intention is zero. There is enough evidence among composers to sub-
stantiate this method of creation as not being uncommon. An intermediate form
is a semi-rational intuitive process, and the extreme, a complete rationalization —
the_engineering of music.

Wi£h the adoption of an engineering technique, the entire approach to mus-
ical patterns becomes mathematical. Scientific analysis of musical composition
rey^al§jtiia.t all the processes involved in the creation of a musical composition
may be represented by elementary mathematical procedures. For a number of
centuries philosophers have suspected that there are unconscious mathematical
procedures behind conscious musical intentions. Music becomes "the mathe-
matics_of-the-soul.'' The raw material of the mathematics of music begins with
atomic structure and the life of living cells. It is quite simple to solve all the
problems of musical creation with the mathematical equipment we possess at
present. All musical procedures are only special cases of the general scheme of
pattern-making. There is even an absolute identity among the series pertaining

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SCIENCE AND ESTHETICS

to the forms of organic growth, to crystal formations, to the ratios of curvature
of the celestial trajectories and orbits, and to the forms of musical rhythm.
Thus we come to the end of the cycle. Music is one of the phenomena of human
experience. The integration of these experiences leads back to the fundamentals.
We learn through music what we learn through astronomy and biology. We
arrive at an idea. Music is one of the embodiments of the idea. In the remote
future of human history through the continuous process of abstraction, this idea
will emancipate itself from its functional associations in the way that a pentacle
emancipated itself from a starfish or a sea urchin. This will be the logical end of
music.

Before music disintegrates, it will acquire greater functional expediency.
It will be manufactured and distributed in the way other industrial products are
manufactured and distributed. Before music disintegrates, it will influence the
allied arts and come into fusion with them. The compound art of primitive man
in his ritual ceremonies develops into individual art forms, which later develop
and acquire their independence. At the end of this evolutionary cycle, the rela-
tion between the allied arts increases anew and they begin at first to influence
each other, and later to fuse with each other. A dance with musical accompani-
ment is one of the most trivial forms of such fusion . Not long ago it was visionary
to admit the fusion of photography with speech and music, which is today com-
monplace entertainment.

The International Exposition in Paris (1937) presented the transformation
of liquid masses, combined with a variation of projected color, and accompanied
by music. This art of luminous fountains merged with music took place once
more at the New York World Fair in 1939. Without overlooking the influence
of musical forms upon the dance, we may note that music has influenced literary
forms as well as painting. The patterns of musical composition take place in the
new art of projected light (lumia). "The music of visible images" (abstract
cinema), a comparatively recent development, calls for a greater precision in
both design and music. The most recent and most successful of the new art
forms is a new realism based on the fusion of the two arts: music and design.
It is mechanical realism as we observe it in animated cartoons. These cartoons
are the end of the cycle, beginning with ancient puppet plays. The art of the
cinema has not yet reached its climax. On the contrary, it is too young to dis-
integrate in the near future; yet the amount of engineering technique employed
in all phases of this art is incomparable with the amount of acoustical engineering
that was necessary during the time of Bach or the amount of paint chemistry
that was necessary in the time of Leonardo da Vinci. Television, being the ulti-
mate achievement of the "engineering" of today, will undoubtedly stimulate
further fusion of existing art-forms.

As physiology becomes a branch of electrical engineering in the study of
brain functioning, esthetics becomes a branch of mathematics.

To sum up the evolutionary groups pertaining to art forms, we offer the
following scheme of morphological zones. These zones may follow each
other chronologically as well as overlap each other, and may differ in different
localities.

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E. History of the Arts in Five Morphological Zones

Zone One. Biological. F 're-Esthetic.

The struggle for existence. Defense reflexes. Tactile orientation. Adapta-
tion to the medium. Automatic self-protection. Automatic self-destruction.
Mimicry. Motor reflexes. Signaling.

Zone Two. Religious. Traditional-Esthetic.

Intentional mimicry. Reproduction. Performance. Magic. Ritual Art.
Incantation. Religious art.

Zone Three. Emotional-Esthetic.

Emotion. Artistic expression of emotions. Self-expression as unconscious
mimicry. Origination of an esthetic idea. Art for art's sake.

Zone Four. Rational-Esthetic.

Growth of esthetic ideas. Rationalizing. Rationalization. Experimenting.
Novel art. Modernism. Experimental art.

Zone Five. Scientific. Post-Esthetic.

Analysis and synthesis of an art product. Scientific experiment. Art with
a scientific goal. Scientifically functioning art. Manufacture, distribution and
consumption of a perfect art product. Fusion of art materials and art forms.
Disintegration of art. Abstraction and liberation of the idea.

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CHAPTER 4

THE PHYSICAL BASIS OF BE A UTY

' I 'HE physical source of the arts as functional groups is the off -phase pair, or
group, of sine-waves. One sine-wave is the limit of simplicity in action. Two
sine-waves with a relatively negligible periodic difference produce beats or inter-
ference (pulse). This pulse is the life element, or the manifestation of life, of
an esthetic entity. Thus, phasic differences, causing instability in wave motion,
are the actual factor controlling esthetic varieties.

This proposition remains equally true whether it is applied to the human
voice or a musical instrument. From an esthetic viewpoint, the quality of sound
largely depends upon the form and frequency of this pulse, which in physical
terms is the ratio produced by the difference between two component waves.
When the frequency of this pulse is too low (below 5 cycles per sec), the impres-
sion gained from tone-quality is that of insufficiency, of retarded life speed.
When the pulse-frequency is normal (5 to 6 cycles per sec), the impression gained
from tone-quality is healthy existence, well-being. With pulse frequencies higher
than normal (above 6 cycles per sec), the impression gained from tone-quality
is of accelerated, precipitated, tense existence.

The passive character of the first quality is due to the effect of unstable
equilibrium, which is not sufficiently defined and seems to be below our biological
rhythms. The normally active character of the second quality is due to the
effect of unstable equilibrium, which seems to synchronize with the biological
rhythms of a healthy body. The over-active character of the third quality is due
to excessive instability, which makes the preservation of the equilibrium some-
what of an effort. It overstimulates our biological rhythms.

These three qualities can be defined respectively as sub-biological, biological
and supra-biological. As the first quality can be properly compared with under-
stimulation, the second, with normal stimulation, and the third, with over-
stimulation, the three qualities correspond to the psychological triad: subnormal-
normal-supernormal. The first quality corresponds to depression, consciousness
of weakness, melancholy, pessimism. The second quality corresponds to normal
existence when well-being is not consciously noticeable. The third quality cor-
responds to joy, consciousness of vigor, heroic urge, overactivity and ecstasy.

As the second quality, axis of psychological equilibrium, is the persistent
status quo, we are aware of its absence (not of its presence) only when we deviate
from it. This sensation is comparable with the status quo of the horizon, which
forms our visual axis. We become conscious of it only when it loses its horizon-
tally. Consider the sight of familiar surroundings when viewed from an in-
clined surface, a boat, a float, a train, or a plane.

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19

The psychological triad is in direct accord (and, perhaps, correspondence)
with the frequencies stimulating visual and auditory perception. The low fre-
quency of red affects us as understimulation; the middle frequencies of yellow,
green and blue — as normal stimulation; and violet — as overstimulation. Com-
pare these color frequencies with the sunset with its overabundance of red, the
midday with a balanced spectrum and the forenoon with its ultra-violet pre-
dominance. In this picture, we get biological as well as photo-chemical evidence
(reaction of the emulsion in color film).

The same is true of sound. Low pitches (low frequencies) produce a quies-
cent, nocturnal, sub-biological, understimulating effect. The middle range, par-
ticularly the one which corresponds to the range of the human voice (approxi-
mately between 64 and 1200 cycles) embraces the psychological range of normal
stimulation. The high frequencies produce an effect of overstimulation, par-
ticularly when abundant with beats. Compare the shrill, ecstatic effects of high
flutes.

• Of course, frequency can be dissociated from intensity only for analytical
purposes. It is the combination of intensity and frequency that forms the actual
stimulus. In view of this, it is interesting to note that the alternation of two
different intensities in rapid succession produces the beat or oscillating effect,
and that the periodicity of intensity beats corresponds in frequency to that of
phasic displacements.

We shall now examine our psychological triad in its correspondence to
optical and acoustical mechanical frequencies, i.e., speeds of projecting motion-
picture films with their sound-tracks or playing phonograph records.

In projecting a motion-picture film at 8 intercepted images per sec, we
obtain an insufficient degree of graduality of transition. The quality of motion
appears to us as subnormal; it manifests itself in discontinuous movements.
Sixteen intercepted images per second give satisfactory continuity of motion and
suggest a nearly normal quality. Present-day 24 images per second produce a
perfectly continuous motion of images if the photographed motion is of moderate
speed. To get still more perfect normal effects from very fast motion, we must
speed up both the photography and the projection. To judge which speed would
appear as normal, we can take first the accelerated photography of 32, 64, 128
and more images per second and project them at 24 frames per second, thus ob-
taining "slow motion." If such slow motion gives perfect graduality of transition
from one image to another (continuity of movement), let us say at 48 fr. per
second, it would mean that normal projection of such a movement must also be at
48 fr. per second. Compare the cannon shots at 128 fr. per second which, being
projected at 24 fr. per second, still do not produce the effect of continuity.

Empirically, however, we gain very little by photographing fast moving
objects at excessive speeds and projecting them back at the same speed, as this
realistic restoration, perfect in itself, does not correct the imperfection of our
vision. Thus, we have two choices: either we see fast moving objects at a
stretched time period, wherein we perceive all the details that we can perceive in
continuity; or we see the actual image of a fast moving object realistically
restored, something we cannot actually see.

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SCIENCE AND ESTHETICS

Standard projection (16 or 24 fr. per second) of images taken in accelerated
motion (at 32, 64, 128 and more fr. per second) produces an effect of the super-
normal and even the supernatural. In the same way, the substandard projec-
tion, 8 or less fr. per sec, of normal motion (i.e., motion which under standard
projection and standard taking would appear continuous) produces a subnormal
and even subnatural effect, due to the fact that the intercepted images do not
form continuity.

The sub-biological speed, brought through lapse-shot cinematography to
normal speed (projected at 16-24 fr. per sec.) , is a source of particular fascination.
In this connection, see the "growth of plants" filmed by various scientific organ-
izations, which seem to present magical effects.

In sonic reproduction, the speed at which the sound is recorded requires a
play-back at the same speed. As continuous images in sound are formed at the
minimum of 16 cycles per sec, which is too low to transfer all components prop-
erly to the coating, and mechanically too difficult to achieve in sufficient uni-
formity, the standard 33.3 r.p.m. recording-reproducing speed is fully satisfac-
tory. But we must -remember that satisfactory results with this relatively low
speed are a comparatively recent achievement. The commercial 78 r.p.m. speed
is still used as a standard on most instruments.

The psychological triad corresponds to the recording and reproducing speeds
in the following way: we obtain subnormal effects by playing at a speed lower
than the recording speed. Under such condition the pulse of tone-quality, i.e.,
the beat-frequency slows down more than twice, which is the ratio between 78
r.p.m. and 33.3 r.p.m. According to our previous analysis, this should result in
sub-biological effects. And it does. The voice of tenor Enrico Caruso, under
such conditions of reproduction, sounds like a cow (particularly when the attack 1
appears on "m" coupled with "oo" or "o" or "u.")

Normal effects are obtained, of course, by reproducing at the speed of re-
cording. Supernormal effects are achieved by speeding up the mechanical fre-
quencies in the reproducing apparatus. Thus, music recorded at 33.3 r.p.m.
and played back at 78 r.p.m. displays supernatural agility and affects us as over-
stimulation .

Subnormal effects bordering on the sub-natural may be obtained by taking
the original record made at 78 r.p.m., playing it at 33.3 r.p.m., re-recording it
at 78 r.p.m. and playing it again at 33.3 r.p.m. In the same way, supernatural
effects may be produced by reversing this operation. A record made at 78 r.p.m.
must be played at 78 r.p.m., re-recorded at 33.3 r.p.m., and played again at 78
r.p.m. This procedure may be continued in both directions until we get no
sound at all.

This zero of sound may be paralleled with the zero of motion that we obtain
in a motion-picture by projecting just one individual frame upon the screen, or
by standard projection of a very slow-moving object, photographed at an in-
credibly high speed. Under the latter condition, the object would appear sta-

'Schlllinger uses "attack" to mean the pro-
duction of sound, in this instance, by means of the
vocal cords. (Ed.)

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THE PHYSICAL BASIS OF BEAUTY

21

tionary, for the phasic changes that may be noticed by the eye would be dis-
sociated by very low time intervals. Compare this with the stationary appear-
ance of a starry sky.

It is important to realize that the ratio of audible frequency-limit equals
1000, whereas the ratio of the wave-length visible spectrum is-r- It may be ex-
tended for convenience to the ratio 2, i.e. ,-§-. The wave-length of extreme violet
is .00040, and that of extreme red .00072. As infra-red has a wave-length of .001 ,
such an extension of range as does not reach infra-red may be fully justified for
empirical purposes of color composition. Thus, ranges do not correspond. For-
mation of perceptible kinetic visible or audible continuity, on the other hand,
does have a common ground. The sequence of sound wave periods is 16 cycles
per sec. minimum. Such a speed of impulses produces the effect of a continuous
sound. The number of intercepted images in cinematic projection is set at 16
per sec. as a minimum producing continuous motion.

Esthetic pleasure grows with the increase of frequencies in both cases. How-
ever, sound and visible image have their own limits. Sixty-four to one hundred
and twenty -eight frames per second are used in "slow motion" cinematography,
which always delights audiences. The newly developed high speed cinematog-
raphy (over a 1000 frames per second) transforms an ordinary phenomenon like
a milk drop into a stirringly beautiful spectacle. In sound, it seems that the
majority of our listeners prefer tones between one hundred and three hundred
cycles per second (cello, male tenor, baritone and female contralto).

Since sense organs react to frequencies and intensities as such, and not to
associative psychological forms and images, esthetic objects are capable of direct
stimulation by number-harmonies and proportions present in the artistic media
(sound, color, etc.)

Conditioned reflexes associated with pleasure and delight grow through re-
peated experiences. This is not to be confused with inherited, unconditioned
reflexes. Thus, cultivation of positive reactions toward some kind of new art
takes time and requires repeated experiences. Most people like either familiar
music or music of a familiar kind.

It is only with the growth of refinement of perception in a certain field of
art that an individual acquires an urge (desire, appetite) for the lesser known
and more intriguing. Consider the delight which certain classes of professional
musicians experience over the music of African cannibals.

The beauty, of art material as a purely physical state may be described as
follows:

1 . beauty of the tone itself

2. beauty of the color itself

3. beauty of any component per se

Beauty of the composition as such would result from harmonic relations of har-
monically developed components.

Under such conditions, any component mav constitute beauty: spatial and
temporal components, for example, may constitute beauty. This is where the
harmony of numbers occurs. But a beautiful art work must have beautiful corn-

Digit

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22

SCIENCE AND ESTHETICS

ponents in beautiful correlation. The resultant of synchronization of all compo-
nents is equivalent to beauty. For instance: a melody which appears to be
beautiful has a harmonic temporal flow (time rhythm), a harmonic relation of the
sequent frequencies (pitch), a harmonic beat frequency combined with harmonic
intensity (let us say 5 to 6 beats per second), and harmonic groups of intensity.

It is often true that the art material as such may overshadow other compo-
nents as well as the composition as a whole. For instance, Caruso singing worth-
less and often stupid music may offer a vocal quality so harmonic that the lis-
tener's attention centers on it and he becomes unaware of other components of
the artistic whole, including Caruso's own appearance.

Other examples are cumulus clouds, sunrise, rainbow, and an art form known
as "lumia" and propagated by Thomas Wilfred. 2 The quality of celestial,
luminous shapeless beauty is exactly in the same esthetic category as the images
of the "clavilux." Some spectators attending the clavilux performances find a
"cosmic touch" in the images created by this instrument, in spite of the great
monotony of the temporal-spatial composition.

The beauty of the material per se, i.e., the beauty of texture, differs only
quantitatively, in terms of perceptible dimensions, from any other form of har-
monic configuration. The componeots of texture are microscopically small as
compared with the structural components of an art work. Though the degree of
textural saturation varies, the aggregation of microscopic configurations (molecu-
lar structure of matter of the continuous periodic impulses producing sound)
is so great that the configurations cannot be individually discriminated in sensory
orientation and are perceived as a homogeneous qualitative whole instead.

'Sec art form No. 8 in Chapter 1, Continuum
of Part II. (Ed.)

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CHAPTER 5

NATURE OF ESTHETIC SYMBOLS

A. Semantics 1

USIC in general — and melody in particular — has been considered, since

time immemorial, a supernatural, magical medium. Many great philoso-
phers in different civilizations have given their attention and directed their
thoughts toward this elusive phenomenon. The more definitions of music you
know, the more you wonder what music really is. It seems to fall into the
category of life itself. It seems to have too many "x's."

People did not know much alx>ut lightning even ten thousand years ago,
and ten millennia make only a one-hundredth in the range of human evolution.
We tend to ascribe supernatural powers to any phenomenon we cannot explain.
Today, we are surrounded by things more miraculous than any of the products
of ancient imagination — and when you think of the achievements of modern
technique, it seems to be incredible that a toy — as simple as melody— should still
remain in the category of the irrational.

Following our method of analysis, however, we may assume that any phe-
nomenon can be interpreted and reconstructed. To accomplish this, it is necessary
to detect all the components and to determine the exact form of their correlation.

There are two sides to the problem of melody: one deals with the sound
wave itself and its physical components and with physiological reactions to it.
The other deals with the structure of melody as a whole, and esthetic reactions
to it.

Further analysis will show this dualism is an illusion and is due to consider-
able quantitative differences. The shore-line of North America, for example,
may be measured in astronomical, or in topographical, or in microscopic values.
The difference between melody from a physical or musical standpoint is a quanti-
tative difference. The differentials of the physical analysis become negligible
values for purposes of musical (esthetic) analysis.

Melody is a complex phenomenon and may be analyzed from various stand-
points. Physically, it can be measured and analyzed from an objective record,
such as a sound track, a phonograph groove, an oscillogram or the like. Melody
when recorded has the appearance of a curve. There are various families of
curves, and the curves of one family have general characteristics. Melodic curve
is a trajectory, i.e., a path left by a moving body or a point. Variation of pitch
in time continuity forms a melodic trajectory.

l This chapter is part of Book IV, Theory of position. Copyright 1941 by Carl Fischer, Inc.
Melody, of The Schillinger System of Musical Com- Reprinted by permission. (Ed.)

UNIVERSITY OF MICHIGAN

24

SCIENCE AND ESTHETICS

Melody from a physical standpoint is a compound trajectory of frequency
and intensity. Melody from a musical viewpoint is a compound trajectory of
pitch, quality, and volume. The components of quality are timbre, attack, and
vibrato.

Physically , pitch is an accelerated periodic attack . Physically , the difference
between rhythm and melody is purely quantitative. Therefore, time-rhythm
in a melody may have two forms.

1. Through periodicity of attacks of low frequency, which is unavoidable
when the pitch-frequency is constant;

2. Through variation of frequencies, i.e., through changes in pitch itself.

Frequency constitutes musical pitch. Any sound wave of a given frequency
(constant or variable) generates its own frequency subcomponents (known as
"partials" or "harmonics") resulting from purely physical causes. The latter
are disturbances which convert a simple wave (known as a sine wave) into a
complex one. The sound of a simple wave may be heard on specially made tuning
forks and electronic musical instruments.

The intensity of a sound wave is one of the factors of disturbance, and the
duration of intensity and its stability »in time continuity are others. The latter
are musical factors: depend on form of attack (or accentuation). Finally, the
resultant of both components and all the subcomponents, i.e., the interaction
of all component frequencies and intensities in a sound wave, constitutes the
musical component of timbre and character (quality) of sound.

The relative importance of musical components and subcomponents has
already been measured, so to speak, by agreement among musicians and music
lovers. The conclusion has been reached that two melodies are identical if their
main components (time and pitch) are identical. For instance, a melody played
on the piano, or sung, or played loud or soft, or with vibrato or without it, would
be considered "the same" melody if rhythm (time) and intonation (pitch) are
identical. The subcomponents and the sub-subcomponents pertain to execution,
i.e., to the performance of melody, not to its own structural actuality. The
very neglect of subcomponents, on the one hand, relieves the composer of a cer-
tain amount of responsibility; on the other hand, it leads to loss in esthetic value
of melody when the melody is wrongly executed by the performer. For then the
performer has to supply the subcomponents without the benefit of any exact
indication by the composer and therefore he acts at his own discretion, whether
rightly or wrongly .

At this point we may adopt Helmholtz' definition of melody (which satisfies
the musical aspect): melody is a variation of pitch in time. 1 Is any variation of
pitch in time a melody? An attempt to answer this question leads into the
semantics of melody.

•Hermann Ludwig Ferdinand von Helmholtz sounds. His most significant work is On the

(1821-1894). the great German physicist and Sensations of Tone as a Physiological Basis for the

physiologist, sought to devise rules of musical Theory of Music published in 1863. (Ed.)
science based on the physical nature of musical

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NATURE OF ESTHETIC SYMBOLS

25

B. Semantics of Melody

The fundamental semantic requirements are that melody must "make
sense," it must have (like words) associative power, i.e., it must be able to con-
vey an idea or mood, to "express something."

But these are also the requirements of language, and yet there is a distinct
difference between word and melody as symbols of expression. The function of
words is to express the concept of actuality , to find its verbal symbol . The function
of melody is to express the structural scheme of actuality. Words have their
origin in thought; melody has its origin in feeling, i.e., originally in the reflexes.
Words generate concepts which may or may not stimulate feelings. Melody, on the
contrary, stimulates feelings (emotions, moods) as spontaneous reactions, which
may or may not generate concepts. Melody expresses actuality before the concept
is formed for that actuality. This is why, in listening to a melody, one is satisfied
with its expression to such an extent that the quest for the concept, "What does
it actually express," is never aroused. But, on the contrary, when a melody does
not convey sufficient associative power (to stimulate reflexes, reactions or moods),
then the listener looks for a verbal description of it, or, at least, for a title, a
"label," a concept. Melody is insufficient whenever it calls for a verbal explana-
tion. When a word does not convince through its own associative power, or in
order to increase the latter, one resorts to intonation and gestures.

Words or melody may or may not be self-sufficient. Words that are not
self-sufficient call for a specific form of intonation in order to acquire the necessary
associative power. We may also state, reciprocally, that melody which is not
self-sufficient as intonational form calls for word and often for a symbol in the
form of a verbal concept. These two statements can be verified by simply study-
ing the facts.

Here we arrive at the idea that although, in their developed forms, both
word and melody are self-sufficient — in their early periods of formation they
produce hybrid forms: an intonational form that calls for a concept — and a
conceptual form that calls for intonation.

Here are a few of many references. According to the statements of George
Herzog, Columbia anthropologist who made some pertinent recordings and
demonstrated the phonograms, there are certain Central African tribes whose
verbal language is just such a hybrid. A word of the same etymological constitu-
tion (spelling) has at least ten different forms of intonation, each attributing a
different meaning to the word. In this case intonation is an idiomatic factor .

In other cases, as in some instances of Chinese music, 3 melody or even the
single units of a scale become symbolic of a concept — i.e., they assume the func-
tion of words.

The Stony Indians of Alberta, Canada, try in their songs to express the
sound of a brook, the murmur of leaves, etc. Yet as a descriptive means it is
not self-sufficient; it calls at least for a title. This is a case in which melody is a
bad competitor of poetry.

•See Karl Stork. History of Music. (J. S.)

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26

SCIENCE AND ESTHETICS

Out of many hypotheses as to the origin of music and word, I select the
reflexological one. 4 Sound reflexes (of the vocal cords) , before they crystallized
into relatively distinguishable forms of word and melody, were spontaneous
expressions of satisfaction or lack of satisfaction in an animal organism. Any
cause of actual or potential disturbance that endangered an organism became a
stimulus for the defensive reflex. This is probably the original form of the in-
tonational signal. If such a form was at first an improvised reflex movement
of the vocal cords expressing fear — a spontaneous reaction to danger — it may
have crystallized later into the etymological form of the concept of "danger."

When an organism is on the verge of struggle for its own survival, it usually
resorts to intonational signaling rather than to an etymological one. Even in
our own time, a drowning man does not say: "I am drowning!" He generally
shouts: "Help!"

The amount of semantic and acoustical elements in words or melodies varies
greatly. There are all gradations from an exclamation to a polyconceptual
polysyllabic word of the German language with the relative decrease of the acous-
tical (intonational) and the relative increase of the semantic element. In many
undeveloped forms of speech, an outsider may in fact mistake such speech for
melody . 5

Melody always contains well-defined acoustical elements, although it may
be alien to an ear trained to different systems of intonation. Melody offers
also a scale of semantic gradations from imitative descriptive intonations,
through symbolic abstractions, to the expression of mechanical forms.

Both imitative and symbolic functions of music tie it closely to verbal
semantics. In this stage, melody is the language of a given community only.
Tests show that even such commonplace moods as "gayety" and "sadness"
cannot be expressed by means of melody that will mean the same thing to all
nations. Melody is a national language or a language of a given epoch with regard
to descriptive or symbolic qualities.

Arabian funeral music sounds anything but "sad" to us because of our
association with major scales — which mean gayety, heroism, happiness and
satisfaction to us. Gay Arabian dance-songs sound "sad" to us because of our
association with harmonic minor scales, which mean exactly the opposite to us.
It is similar with the forms of musical harmony. Through previous associations
we react to major chords as we react to major scales. Yet we have the curious
phenomenon of the Negro- American "blues," which is supposed to express de-
pression, but which, nevertheless, has the richest scale of major chords.

All the controversies ascribing this or that semantic connotation (descriptive
or symbolic) to music will vanish when we penetrate the real meaning of music,
namely, the expression of the forms of movement. The objectification of this

4 Here begins, in a partial form, Schillinger's
exposition of his theory of the correspondences
between music — melody, in particular — and the
objective world of life. As such, his theory offers
us the means whereby esthetic phenomena can
be correlated in a scientific and materialistic way
with the rest of human experience; in consequence.

Digitized byGoOgle

even this partial exposition is of the utmost philo-
sophical importance. (Ed.)

s "Program music is a curious hybrid, that is,
music posing as an unsatisfactory kind of poetry."
— Oxford History of Music, Volume 6, Page 3.

(J. S.)

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NATURE OF ESTHETIC SYMBOLS

27

meaning requires only one premise: biomechanical , physiological experience , com-
bined with a highly developed sensory system. The requirement may be satisfied
by any normal specimen of the higher.animal forms. 6

Though commonly unknown and generally repudiated when brought into
a discussion, this fundamental form of musical semantics had already been
known to Aristotle. Here is his definition: "Rhythms and melodious sequences
are movements quite as much as they are actions." This is the first historical
instance of penetration into the true nature of musical language.

The meaning of music evolves in terms of physico-physiological correspon-
dences. These correspondences are quantitative and the quantities express form.
This can be easily illustrated by the following example.

A sound of constant frequency and intensity and made up of a simple wave
affects the eardrum and the hearing centers of the brain as an excitor of a simple
pattern. Such a pattern may be projected by various means so that its structure
becomes apparent to another more developed, and therefore more critical organ
of sensation, that of sight. The complexity of reaction {i.e., its form) is equivalent
to the complexity of the form of the excitor. The number of components in a
wave affects a corresponding number of the arches of the inner ear's Cortis organ,
putting them into oscillatory motion. If a sine wave has one component, it
will affect only that arch which reacts on the frequency corresponding to that
transmitted through the air medium in the form of periodic compressions. When
a wave of greater complexity affects the same organ, the reaction becomes more
complicated .

It is a known fact that the ear can be trained. Therefore, the pattern of
reaction is equivalent to the pattern of excitation with various degrees of ap-
proximation. All the components of sound work in similar patterns because
these patterns are similar in all sensory experiences. Formation of the patterns
is due to (1) configuration and (2) periodicity. The configuration may be simple
or complicated in a mathematical sense, i.e., its simplicity or complexity can
be expressed in terms of components and their relations. This emphasizes both
frequency and intensity in a sound wave, as well as the character of sound which
is the resultant of the relations of the two components. Periodicity defines the
form of recurrence and may be also of different degrees of complexity — for ex-
ample, the periodicity of recurring monomials as compared to the periodicity of
permutable groups.

Our physiological experience, combined with our awareness of that experience
through our sensory and mental apparatus, makes it possible for us to under-
stand the meaning of music in terms of "actions." Thus, regularity means sta-
bility, and simplicity means relaxation. Thus, the satisfied organism at rest is
comparable to simple harmonic motion . The loss of stability is caused by power-
ful excitors affecting the very existence of the organism. Sex and danger are the
excitors, and love and fear are the expressions of instability.

The awareness of such instability comes through variations in blood circu-

'Compare Plato's ideas on the meaning of menu with the pitch discrimination of dogs in
music in his Republic and Ivan Pavlov's expert- his Conditioned Reflexes. (J. S.)

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28

SCIENCE AND ESTHETICS

lation sensed through the heart-beat and variations in blood-pressure, resulting
in respiratory movements. The whole existence of an organism is a variation of
degrees of stability, fluctuating between certain extremes of restfulness and rest-
lessness. The constitution of melody is equivalent to that of an organism. It is
a variation of stability in frequency and intensity. Melody expresses those
actions we know and feel through our very existence in terms of sound waves.

C. Intentional Biomechanical Processes

We come now to a consideration of intentional biomechanical processes.
Efficiency of action in relation to its goal is the foundation of evolution. The
forms of action by which living organisms adapt themselves to the goal of sur-
vival in the existing medium may serve as a fundamental illustration. This
efficiency comes about through "instinct" among the lower species, but through
the conscious utilization of previous experiences leading to deliberate efficiency
among the higher animals. Muscular tension and relaxation constitute the first
instruments of such intentional action.

The mechanical constitution of melody varies with times and places, yet
its patterns are familiar to us from our own biomechanical experiences.

The "contemplative" and the "dramatic" become two poles of our esthetic
reactions. They grow out of the same biomechanical diads: restfulness-restless-
ness, and stability-instability.

Dramatic patterns themselves evolve out of two sources: the first is fear
(defense — dispersed energy) and is caused by danger or aggression; it results in
contraction patterns. The second is aggression (attack — concentrated energy)
and is caused by an impulse or resistance; it results in expansion patterns. Con-
fusion of patterns of compression with those of expansion (aberration of percep-
tion caused by instability) explains why the very same music sounds "passionate"
to one listener but "weary" to another. This is a typical confusion observed
by Professor Douglas Moore of Columbia in tests performed on students of non-
musical departments at various universities, using Wagner's Isolde's Love-Death
as material .

All the technical specifications for melodic pattern-making will be given
later. The immediate question is: how does it happen that the physiological pat-
terns are identical with the esthetic patterns? We can answer this question only
hypothetically for we know very little about the technique of pattern formation
at present. But as science progresses, we notice more and more correspondences
in different fields. We find identical series in such seemingly remote fields as
crystal formation, ratios of curvatures in the celestial trajectories, musical
rhythms, design patterns, and, finally, in the very molecular structure of matter
itself. Modern chemistry shows how by geometrical variation of mutual positions
of the same group of electrons, entirely different substances are produced. Little
as we know for the present about the electro-chemistry of brain-functioning,
we may well suspect that all our pattern conception and pattern-making are
merely the geometrical projection of electro-chemical processes, in the making,
that occur in our brain. This geometrical projection is thought itself.

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NATURE OF ESTHETIC SYMBOLS

29

D. Definition of Melody

The summary definition of melody:

(1) Physiological definition: Melody is an excitor existing in the form of a
sound-wave which affects the organ of hearing. The latter being a re-
ceiver and a transmitter transfers it to the biomechanical pattern-
making center of the brain.

(2) Semantic definition: Melody is an expression of biomechanical experi-
ences in the sound medium.

(3) Musical definition: Melody is a variation of pitch in time, wherein pitch
units follow a preselected scale of frequencies and express a relative
stability of each individual unit.

The summary definition of word:

(1) Physiological definition: Word is an excitor existing in the form of a
sound-wave which affects the organ of hearing. The latter, a receiver
and a transmitter, transfers it to the concept-making center of the brain.

(2) Semantic definition: Word is an abstraction of biomechanical experiences
in the sound medium. "Poetic image" is a variation of the original
biomechanical abstraction .

(3) Musical {tonal) definition: Word is a variation of pitch in time, wherein
pitch units express a relative instability of each individual unit and do
not necessarily follow a preselected scale of frequencies.

It follows from these definitions: (1) that in symbolic notation (though
different patterns are used) — printed letters or musical notes — both word and
melody are identical; (2) a poem recited in a foreign or unknown language be-
comes an undeveloped form of music.

Original from
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"Esthetic perception" is a tautology, as "alsthetikos"
means "perceptive." For this reason esthetic perception
must be defined as a special form of selective perception.

Selective (esthetic) perception is a capacity to discrimi-
nate relationships through senses and to associate such
inter r elatedness with the functionality of structure.

CHAPTER 6
CREATION AND CRITERIA OF ART

A. Engineering vs. Spontaneous Creation

"\AATHEMATICAL Basis of the Arts is a scientific theory of art production.

It classifies all the arts according to the organs of sensation through which
they are perceived: sight, hearing, touch, smell and taste. It usually takes the
lifetime of a genius to make a sizable contribution to any art, although some
artists have attempted to paint, to compose music and to become mechanical
inventors at the same time. And though a complete mastery of more than one
art has been attributed to very few, the N problem of scientifically coordinating
several arts in one has never been accomplished by one individual.

The scientific theory of art production approaches this problem and solves
it exactly by the same method as the problem of locomotion on the ground, in
the water, and in the air is solved — that is, by engineering. The argument of
spontaneous creation must be repudiated, particularly since works of art gener-
ally conceded to be among the greatest, have not been produced spontaneously.
Quite on the contrary: the process of creation consumed an enormous amount of
time and considerable mental and emotional effort.

The difficulty with spontaneous creation is due to the fact that an organic
work of art is a combination of various components. For example, when a com-
poser creates a theme, such a theme implies the co-existence of melody, rhythm,
harmony, dynamics, phrasing, etc., all in one. A scientific approach first de-
velops each component individually, then assembles them into a definite coordi-
nated whole.

A building is not erected by magic. First comes the architect's idea. This
idea is based in part on existing material forms and their properties. After the
architect makes a blueprint and the contractor is called upon, only then does
materialization of the idea begin. Excavation of the ground, draining, cement-
ing, erection of steel girders, installation of the plumbing system, electric wiring,
plastering, painting — all these phases of construction, following each other in a
pre-determined manner, bring us finally to such achievements of contemporary
engineering as the Empire State Building, the Golden Gate Bridge, and other
important structures. A spontaneous creation in the field of architecture would

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CREATION AND CRITERIA OF ART

31

probably result in nothing more complex than a log cabin. Compare the amount
of engineering involved in an African canoe carved out of one log or trunk, and
that necessary in the construction of a modern battleship.

Music, as well as the other arts, still relies upon cave-age methods of pro-
duction. One may build spontaneously and without computation a simple hut,
a cave, a tent, or a cabin; but spontaneous creation of a skyscraper would only
result in disaster. Even inexpert engineering could not be relied upon to
carry out such a task. When some Parisian company tried in 1928 to erect its
first skyscraper following the American trend, the structure fell after the steel
had been erected to the ninth floor. And that was in Paris, where the Eiffel
Tower stands as a symbol of engineering genius. It is clear that one art form is
complex enough to handle. Several arts cannot be integrated without the aid
of engineering.

An art structure in its formation (the process of being created) may be ex-
pressed as a series of sub-structures in their sequent development and accumu-
lation.

The difference between engineering and the artistic method of production
is mainly that in the scientific method, each sub-structure or component is de-
veloped individually and correlated thereafter with the other individually de-
veloped components. In the artistic process all (or nearly all) components
appear simultaneously as an a priori coordinated group. Any change in such
a group, with the intent of perfecting one of the components, changes the balance-
ratio of the entire group, thus necessitating laborious reconstruction of other
components. The latter frequently changes the balance-ratio so completely
that the final product only remotely resembles the originally intended structure.
With the scientific process, on the contrary, the development of individual com-
ponents may be carried out to the utmost perfection. At the same time their
relationship with other components of the same structure may be constantly
controlled and integrated .

Thus, in the scientific process, there are to be found all the consecutive points
of the series through which individual components are correlated with other
components. In the artistic process, in contrast, many terms of the series are
missing. Moreover, groups of components are generated simultaneously, which
lack logical and esthetic coherence with the final form of the product.

When the artist begins his process with an individual component (such as
rhythmic pattern in music or linear configuration in design), the image of such a
component frequently lacks clear definition. The process may be compared with
seeing an off-focus image and gradually focusing it. Imagine, for example, a
pentagon gradually transforming into a hexagon. Such transformation corre-

S>=S I ti + (S I +S„)t 2 + (S I H-S II ^S III )t a + . . .

I ) . . . + (S, S„ -r S,„ + . . . 4- S N ) t n

S = sub-structure

tii tj, t a = consecutive moments in time

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SCIENCE AND ESTHETICS

sponds to a modulation from 180° to 120° uniform continuous motion. 1 All inter-
mediate values between 180° and 120° do not produce closed polygons. Now
imagine a diffused polygon (i.e., one with an infinite number of sides) gradually
approaching the 120° form and, finally, attaining it. This is the foundation
from which a mathematical definition of the creative process may be deduced
quantitatively. If the desired esthetic form is a regular hexagon, for example,
the creative process can be defined as increase or decrease of angular values ap-
proaching the 120° limit under uniform periodic motion.

In the arts, the final product often does not reach its limit (focus) but re-
mains in a partially diffused state. Definition of the degree of diffusion becomes
an esthetic factor in such a case. In short, the degree of diffusion itself becomes
an art component, as in the paintings of Seurat and Pissarro, and in music
generally, where "un poco piu mosso" or "piano, crescendo, forte" only suggest
variable limits.

B. Nature of Organic Art

The significance of art can be measured by its immediate appeal and the
effect of naturalness. True art, which can be defined as natural and organic art,
has a general appeal and does not require any explanations, just as birds' plum-
age, their singing, the murmur of a brook, leaves, mountains, glaciers, water-
falls, sunsets and sunrises do not require any explanation . The art "isms," despite
lectures, commentaries and volumes of analysis, do not become a jot more appeal-
ing. Despite the propaganda, these "isms" cultivate little more than hypocrisy
in the semi-literate and pseudo-cultured strata of the population.

What is the criterion of a natural and organic art? It possesses character-
istics not present in the art "isms." These are coherence of structure which enables
it to survive, and high associative (semantic) potential which results from such
coherence. For example, a well-defined and economically expressed thought, or
idea, has greater persuasive power than one which is vague and incoherently ex-
pressed .

Art is organic when its form can be traced back to its organic source, as the
winding staircase can be traced back to antlers, horns and cockleshells. The
ancient Egyptians and Greeks discovered the forms of organic art through the
principle of "dynamic symmetry." This principle was based on a so-called sum-
mation series; in this case, the series in which every third number is the sum of
the preceding two. This series, known to the Egyptians and Greeks, was brought
to the attention of the American public in 1920 by Jay Hambidge in Dynamic
Symmetry. Known through the ages, it was disclosed in Luca Pacioli's treatise,
De Divina Proporzione. Leonardo da Vinci, Michelangelo and many others
adhered to it in their creative work. Today, it is being applied to many things
in everyday use, including book jackets and radio cabinets, as well as in still-life
paintings, landscapes and portrait work by contemporary painters. Credit for

'The details of such transformation are to be
found in Chapter 2, Production of Design, of Part
III. (Ed.)

Digi]

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33

the mathematical formulation of this series goes to Fibonacci* (13th century)
and the series itself bears his name.

It is the organic structural constitution of a pitch or color scheme that reveals
itself as esthetic harmony to our ear, eye, or any other organ of sensation. If
we had the olfactory organs of a dog, we would probably now possess an art of
smell comparable with the art of music.

Experimental artists, the "ism" makers like Picasso, Kandinsky, Klee, not
to mention hundreds of their followers, are not bringing us a step closer to the
evaluation of organic art forms. Many of these "modern" artists merely impose
upon us their unsound, insane, feverish, dreamy and distorted hallucinations of
an actuality given to us by nature to be consumed and enjoyed, together with
oxygen, the ultra-violet rays of the sun, the energy of the ocean, and the smell
of the pine forests.

What gives us the right to call these highly imaginative works of art "in-
sane"? The same criteria which would bring about a similar diagnosis, if we
could submit the art products of these artists to the examination of an expert
psychiatrist. But do we want to be driven to insanity? Is this the function of
the arts? When I attended an epoch-making exhibit of Dada, Surrealism and
Fantastic Art held at the New York Museum of Modern Art in 1938, I could
not help thinking how unfair the Museum was to the poor souls held in various
insane asylums, who often project equally as disturbing products of their imagi-
nation. When I reached the fourth floor, my call for justice was answered by
that part of the exhibit, which was actually contributed by the inmates of insane
asylums.

It is often believed that the greatness of a work of art lies in its persuasive
realism. You look at a painting, and it just lives. And there are paintings that
are true to life. But what is such realism save a mimetic reproduction of re-
flexes? The facial and figure expression associated with suffering appears on a
canvas, and the critics praise it. Why? Look around, or go to the places where
misery flourishes and you will see more than you care to. But isn't it ethically
superior to suffer without outward expression? But then artists wouldn't have
anything to paint. If painting were confined to slavish realistic mimicry of
outward expressions, works like "Ivan the Terrible Killing His Son" by Repin
would be among the greatest paintings, as in this composition blood stains are
truly horrifying in their true- to-life quality. Yet to see real blood stains and
witness a real scene of assassination is still much more horrifying. In the age
of improved color cinema we can get not only a realistic record of such a scene,
but we can make it still much more horrible by extending the torture in the film
itself, i.e., by the technique of "slow motion."

On the other hand , we hear and read that the art of Picasso is very important.
Why? Apologists say because he has found a new way of expression . Is it really
new? The idea of giving two heads to one rooster is very childish, particularly
when you recall what the ancient Greeks did to a Hydra, and the Hindus to their

'Leonard of Pisa, who developed the additive nacci) , which resulted in his being known as Fibo-
series in 1202. was the son of Bonaccio (filius Bo- nacci, the name of the series. (Ed.)

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SCIENCE AND ESTHETICS

own God Shiva. Of course, we can be referred to the technique by which this
has been accomplished. But then it leads us back to kindergarten craft.

There should be one criterion in art and art appreciation: its sanity as re-
vealed in its organic quality, that is, its life, and ultimately — growth. When
artists try to be original either by robbing the kindergarten world of ideas or by
projecting nightmares, it means that art as expressed by such artists has reached
its dead end . The constant rejuvenation of art comes through the modernization
of old and even ancient folk traditions; but one has to remember that these
ancient art traditions were closely associated with natural forms and resulted in
truly significant art, whereas the revised versions of it, usually with the prefix
"neo," are merely degenerating traditions. There is no progress for art in the
revision of old forms, that had their place, significance, and became monuments
of history, as in the ancient Egyptian, Greek, Hindu, Javanese, Chinese, and
other civilizations. Let us leave the foolish and meaningless productions of
hybrids to the milliners, dressmakers, and other "creators of fashion."

C. Creation vs. Imitation

When artists, thirsty for the primeval source of art, imitate the art of ex-
tinct civilizations, they are on the wrong track. The true "elixir of life," which
is progress itself, is never in imitation but in creation. Let us see how creation
has revealed itself in the great civilizations of the past.

The difference between the creative and the imitative processes of art pro-
duction lies mainly in the difference between projecting forms from a given set
of principles, whether consciously or unconsciously used, into a given artistic
medium, and merely imitating the appearance of such forms in that medium.
In primitive ornamental Aztec design, the underlying principle arises from recti-
linear, rectangular and diagonal elements. By means of these elements, Aztec
artists accomplished the projection of human and animal forms. The ancient
Egyptians and Greeks used, as an underlying structural principle, the equality
of ratios which can be expressed by means of the summation series, and which
the Greeks called symmetry. 8

Greek sculptors did not try to make a figure of a deity through copying the
appearance of some living model but by establishing a system of proportions
a priori. Such regulative sets, as the set of proportionate relations in this case,
or the rectilinear elements of the Aztecs, constitute limitations which make art
what it is, i.e., a system of symbols integrated in a harmonic whole. Thus, the
Egyptians, Greeks, and Hindus established standards of beauty which were
expressed through the symbols of harmonic relations.

The same is true also of more primitive civilizations. In some forms, esthetic
expression derives from direct reproduction of the surrounding medium — as in the
matching of colors typical of the surrounding sky, vegetation, or birds' plumage.

*The Greek concept of symmetry obviously had try as a correlation of proportions; hence the use

little in common with the modern conception of of the word "dynamic" to distinguish it from the

symmetry as the balanced distribution of elements contemporary "static" concept. (Ed.)
around an axis. The Greeks conceived of symme-

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CREATION AND CRITERIA OF ART

35

Some of the media are directly supplied by the natural resources of the immediate
vicinity, such as clay in Indian pottery. Nevertheless, the transformation of
natural resources, as in the treatment of clay to render it black, or the invention
of a simplified counterpart of the surrounding impressions (esthetic symbols),
are in the domain of true creation.

Imitative art, on the other hand, has no underlying principles, except the
reproduction of appearances as close to the appearance of the original as can be
achieved by manual craftsmanship. In this respect, any scientific apparatus,
either optical or acoustical, gives an infinitely more accurate reproduction of the
visible or the audible image.

The same can be said about the acoustical perfection of present-day sound
recording whether on coated discs, tape or film. The real difference between
the art of painting and color photography, when "true to life" reproduction of
the image is the goal, is purely quantitative. Painting requires the use of a
brush, and no brush can have a point as small as the size of a grain, or a chemical
molecule in photographic emulsion. One of the great artists of all time, Georges
Seurat, reduced the optical elements of an image to the extreme limit possible
with the use of a brush. He built his images with miniature spectral points.
This art became known as "pointillism," "dotting" in literal translation. It
evolved its images from the elements, which consisted of a material point of the
spectrum. The final image was integrated by means of points of different lumi-
nosity. This approach, very close in idea to the principle of televised images,
is far closer to nature than any artificial conception of structure and its elements.
Breaking down an image into some system of elements, in such coordination
that they reconstruct the image as an independent counterpart of the original,
constitutes a true art .

However, neither brush technique nor photochemical synthesis in them-
selves constitute art in the esthetic sense. Nor can craftsmanship alone produce
an original work of art. In order to attain the latter, the system of elements
participating in the building of an image must be developed into a harmonic
whole. Such a harmonic whole is the complex of harmonic groups of various
orders. Proportions pertaining to space occupied by the created image; harmonic
relations constituting the arrangement of elements in the image itself; correlation
of the harmonic relations of the image with the harmonic relations of the space
occupied by the image; components of color, such as hue, luminosity, saturation
brought into harmonic relations with each other and correlated with the space
and the image, implying both color harmony and quantitative spatial relations —
all these taken together constitute an artistic whole that is harmonic. With this
in view, color photography, which no doubt will be highly perfected as a technique
in the near future, should not be considered an art inferior to painting. On the
contrary, the elements of the latter are incomparably cruder, and therefore
not capable of the refinement of expression possible with the microscopic
elements.

In the artistic accomplishments of the future, where the problem of com-
position itself will be based on a refinement of expression worthy to be a counter-
part of our complex psycho-physiological organism, the refinement of the medium

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SCIENCE AND ESTHETICS

will be a necessary asset. Artists will have to forego thinking in terms of imita-
tion, and even in terms of creation based on the primitive systems of the great
extinct civilizations of the past. They will be compelled to devise a creative
system, which, in its refinement, exactitude and complexity, will be an adequate
counterpart of the efficiency of present-day transmitting media, such as sound-
recording, cinema, and television.

It is unfortunate that most artists and esthetes do not realize that the future
of the visual arts lies not in the improvement of painting but in the development
of kinetic visual forms, the crudest of which we succeeded in developing during
the last thirty years. In spite of its youth as an art form, the motion picture
already has become the dominant form of entertainment. Now, with the advent
of color, it has already achieved outstanding works of art, which could not have
been attained in any of the traditional art forms in a comparably short period.

Every art form we know or can imagine today was at one time or other a
magical procedure. Whether in the form of symbolic movements, ritual dances,
symbolic traces on sand, as in the Navajo sand paintings, or the sounding magic
procedures (such as noisemaking and incantations) , magic art always had a for-
mula. In his primitive life, man learned that nature, as well as animals and
human beings, could be acted upon by definite formulae. This notion has de-
veloped during late centuries into scientific formulae such as are used in chemistry
or engineering. When we combine chemical elements in order to obtain a definite
reaction, we use a formula which takes into account the molecular structure of
the elements and their effects upon one another. This analysis discloses why
some numbers and their relations have been believed to possess magical power.
A stimulus of a certain kind , when applied , necessitates a certain kind of reaction ,
depending on the reacting entity and its response to a particular stimulus. In the
past, magic formulae crystallized through numerous repeated experiences. Basi-
cally, there is no difference between the use of a magic formula and a scientific
formula, except that the first is judged by the result of its use while the second,
by all the functional relations of the interaction between the elements of the
stimulus and the reaction.

There is a long evolutionary chain, virtually without demarcation lines,
that has been established from physical and chemical reactions of so-called "in-
organic" nature; through the biological, that is physico-physiological reactions
of so-called "organic" nature; to reflexes and complex reflexes constituting in-
stinct; and, ultimately, to responses known as associations. Depending on the
state of development of the given substance, its reaction may be considered purely
biological, as in the protective coloration of a chameleon; or artistic, as in the
associative color groupings in the works of Kandinsky. They may range from
the reflexes of muscular contraction of an animal overwhelmed by fear to the pro-
jections of diversified horror, "inspired" by uncoordinated associated processes,
as in the dream projections of such experiences in the paintings of Dali.

One of the most remarkable consequences of primitive man's experiences,
which placed him far above the animal world, was that effect is caused by the
extensive repetition of identical or analogous processes. The first outstanding
result of this knowledge was that fire results from rubbing wooden surfaces to-

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CREATION AND CRITERIA OF ART

37

gether. This information alone, through associative memory, led to the use of
tools for carving, sewing, etc.

With the advent of science, it was discovered that the repetitious application
of waves in motion may affect matter itself and be of constructive as well as de-
structive power. Physicists found that supersonic waves decomposed sugar.
The "magic" of sympathetic vibration noted among the phenomena of wave
motion is merely a different form of magic formulae. And if we look more ana-
lytically into the source of various stimuli producing both physical and psycho-
logical effects, we find that they spring in the end from the same source, i.e.,
chemical reactions and electron arrangements in the molecules. In short, we
discover that the fundamental aspect of all stimuli is one or another form of
regularity of occurrence, i.e., periodic motion.

It may be stated that periodic motion, usually known as "rhythm," and
used as a term in the arts as well as in medicine and astronomy, is the foundation
of all happenings in the world we know. This has been sensed from time im-
memorial. The character of rhythm, as its implications grew, has become more
and more mysterious, as nobody has succeeded in analyzing it. In truth, no
scientific study has gone beyond the elementary forms of regularity encountered
in physics and chemistry.

The fact that artistic intuition frequently functions in primitive as well as
in developed art, according to some mathematically conceivable regularity,
emphasizes the point. If we recognize that this is so, it is clear that we must
know more about the nature and forms of this regularity. The main branch of
this book, Theory of Regularity and Coordination * is devoted to the solution of
this very problem. Since the solution has been found, questions pertaining to
the origin and evolution and manifestation of the arts can be answered directly.

'See Part II and Appendix A.

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CHAPTER 7

MA THEMA TICS A ND ART
A. Uniformity and Primary Selective Systems 1

'"THE foundation of this system is the concept of uniformity. This concept
may be evolved from or associated with various axiomatic propositions.
The most basic of the latter are:

1. The system of count based on the so-called natural integers (1 , 2, 3, . . .).
where the addition of the constant minimal unit makes it appear funda-
mental;

2. The system of the measurement of space, where all minimal units form-
ing the scale of measurement are equidistant and correspond to the set
of natural integers;

3. The clock system of time measurement, based on the sequence of uni-
form time instants and corresponding to the set of natural integers.

These three forms of uniformity appear to be axiomatic in their respective
fields. The first, in the field of reasoning, i.e., logical association. The second,
in the field of sight, i.e., perceptive visual association. And the third, in the field
of hearing, i.e., perceptive auditory association as aroused by the ticking mechan-
ism of a clock, a metronome, or of any mechanism producing audible attacks at
uniform time intervals; also, in the field of vision, i.e., where motion in space
follows uniform intervals in time, and where each consecutive phase of such
motion, or the spatial interval between the two symmetrically arranged positions,
is immediately apparent to sight. Such is the case of a pendulum, where the two
extreme positions are apparently dissociated by space and time intervals. These
space and time intervals appear to our consciousness to be in axiomatic, i.e.,
one-to-one, synchronized correspondence.

There are other correspondences which, while elementary, are not axiomatic.
These take place when the relationship between the optical and acoustical images
is more complex. When a wave of one component, i.e., of simple harmonic
motion, such as the tuning fork, is projected on the screen of an oscillograph
by means of the cathode-ray tube, the visual image appears to be axiomatic,
due to the recurrence of the phase pattern; but the sounding image, being con-
tinuous and of relatively high frequency does not disclose its periodicity to the
sense of hearing, because the individual phases of oscillation are too fast for
auditory discrimination. The usual synchronization-controls of an oscillograph

'This chapter represents the most complete and summarizing the underlying ideas, it offers a corn-
most compact statement available of the founda- prehensive survey of the material presented in
tions and rationale of Schillinger's "Theory of Part II of The Mathematical Basis of the Arts.
Regularity and Coordination." In addition to (Ed.) Original TTQrff

38 UNIVERSITY OF MICHIGAN

MATHEMATICS AND ART

39

produce a fairly stationary image on the screen of even two phases, in which
case the regularity can be observed immediately.

Finally we arrive at more complex forms of dependence in the field of uni-
formity. Such is the logarithmic dependence of ratios within a given limit-
ratio. For instance, within the limit of -§• ratio we can establish a scale of uni-
form n ratios. These correspond graphically, i.e., geometrically, to equidistant
symmetric points, which can be represented on a straight line, the extension of
which would correspond in frequencies to the ratio of-|-. In such a case, the
first point of the linear extension corresponds to b and the last point, to a. All
the intermediate points of uniform symmetry become:

We shall consider such sets of uniform ratios as primary selective systems. 1
We shall also make a note that the logarithms of the uniform ratios, uniformly
distributed, become sets of natural integers and are translatable into equidistant
spatial extensions. Thus, we acquire a unified system where ratios correspond
to space-time units.

The refinement of primary selective systems depends on the discriminatory
capacities of perception . The fineness of unit in a selective system is in direct
correspondence to the potential plasticity of expression. A drawing of a human
face with many lines is potentially more plastic than a drawing of the same face
with a few lines. In music, where primary selective systems are tuning-scales .
greater expressiveness is made possible by a greater number of units. A tuning
s ystem of five units, like one of the Javanese scales, is less flexible as a medium for
constructing musical images than some of the Hindu scales where the number of
units re aches 22.

~ Space and time constitute a continuum which is expressible in terms of
physico-mathematical dependence. The continuum of space and time, or the
"space-time" continuum, corresponds to the "dense set" of number values,
which includes all "real" numbers, i.e., both rational and irrational. The geo-
metrical, i.e., spatial continuum, possesses the property of dense sets, meaning
that the number of points in a straight-line segment (which is finite in itself) is
infinite. It follows from the above that primary selective systems are not dense sets.

On the other hand, psychological perception and consciousness, under all
conditions established as normal, form a psycho-physiological continuum, i.e.,
the perception of space-time is continuous .

Y Space-time produces the constant background in which our rational orienta-
tion is capable of discriminating the relations of isolated and group-phases.
These isolated and group-phases in their interaction produce configurations which
may or may not contain any perceptible harmonic relations, i.e., using a different
terminology, "rhythm ." Mathematically speaking, the continuum of space-

t ime perception is an integral of many variables, constituting the manilold7>f
impressions. |

vf, VWP, VW Vf = f

'See Chapter 1 of Part III. (Ed.)

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SCIENCE AND ESTHETICS

As mentioned before, the density of a primary selective series implies a
corresponding degree of refinement. However, any selectiv e series or a set
reaches a point of saturation, not in t he mathematical sense, but in the sense of
perception and rational orientation, ^ eyond this point, the refinement becomes
empirically useless and meaningless. Mach's tuning scale of 720 units per octave
may serve as an example of such a set, as such small intervals do not permit
the auditory capacity, as we know it, to discriminate either configurations or
ratios of the intervals, or any distinctly noticeable difference between the adja-
cent units of the scaled

For this reason, the density of se t, even in view of ultimate refinement as a
goal, is largely conditioned by the discriminatory capacity of the respective
sense-organ, or the degree of coordination necessary between the respective
s ense-o rgans.

As sight is a more developed form of sensation and orientation, it allows the
construction of primary selective systems that are denser sets than in the case
of auditory orientation. It should be kept in mind, however, that any capacity
can be considerably developed by training, particularly if we think of such
capacity as pertaining to discrimination and not to the integral of perception.
For example, a person can be trained to discriminate, and as a result of such
discrimination, to enjoy finer relations in the configurations of pitches or hues;
nevertheless, no lifetime training can augment the individual's perceptive range
of audible frequencies or spectral wave-lengths. True, we do extend the spectral
range to ultra-violet, but this is due to method and not to the growth of the
range of perception .

Summing up this reasoning, we arrive at the following conclusion: discrimi-
native orientation is cap able _of detecting uniformity in the form of perceptible,
discontinuous phases, and configurations in the form of harmonic groups prod uced
by the phases as unit s. This orientation is coexistent with the continuity of the
integral of space-time perception and orientation .

Xhere are two requirements to be added : [first . that the configurations de-
velop within the perceptive and discriminatory range : and secondly, that the
phase-units, i.e., scale, units are great enough for the respective perceptive and
d iscriminatory capacity TJ

The first requirement implies that perceptive range may be different from
discriminatory range. For instance, the perception of pitch-frequencies as a
range exceeds the capacity of pitch-discrimination. It is difficult to specify pitch-
relations, even for a highly trained professional either in acoustics or music,
beyond 5000 cycles per second . Yet the same individual can hear a sound pro-
duced by a frequency four times as great. This means that the discriminatory
range may be two octaves shorter than the perceptive range.

The second requirement implies that no selective scale should be so dense
a set as to approach the perceptible limit of continuity. One consideration must
be added to the last point. In some arts the configurations are so fluent that the
transitions between points representing the members of the selective set, i.er, the
fixed points, are continuous. What makes the fixed points stand out, is the fact
that such points are relatively more stable than the intermediate points of the

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dense set, i.e., of the continuity. The stability itself is expressed by means of
intensity or duration-stress (accent or greater time-value) . This is the character-
istic trait in the execution of Chinese music. When the stability of fixed points
is not sufficient, the configuration may merge with the continuum. This is what
occurs to the sound image produced by a moderate, continuous self-overlapping
surf. Thus, all primary selective scales in this system are based on the uniform
distribution of the range of a component.

Soace-time is considered a general component and is treated as continuity .
All other components, which are perceived by specialized capacities, such as the
visual, auditory, olfactory, gustatory and tactile, are considered special compo-
nents and are treated as discontinuity.

B. Harmonic Relations and Harmonic Coordination

There are other more complex forms of regularity than uniformity. They
are the outcome of combined forms of uniformity, obtained through superim-
position of phases. From the physical viewpoint such forms represent periodicity
of complex phases, i.e., configurations or groups consisting of several simple com-
ponents. Uniformity may thus be regarded merely as a special case of periodic
r egularity, which may be either simple or complex. Forms of regularity pro-
duced by the periodic. recurrence of binomial or polynomial groups are subjected
to detailed analysis in this system. 8

The main theory of the origin of all configurations that are not axiom atic is
base d on harmonic relations and harmonic coordination, and explains in detail
t h~e~source and the technique of composition. This branch makes possible the
masteryjg f "rhythm" as the basis of composition, and establishes its own validity
a s_the system_of _pattern-ma king and pattern-coordinating . It is called Theory
of_ Regularity and Coordinatio n, and in the not too distant future should have
repercussions in all fields of scientific investigation embracing the liberal, the fine
and the technical arts.

Application of this theory to music and visual arts has already produced
far-reaching results. Facts demonstrate and prove beyond doubt that all efforts
in the field of art-evolution, whether intentional or not, are the expressions of
certain tendencies toward a certain goal. This theory can define the goal by
detecting the tendencies as they reveal themselves in the course of evolution.
It means that without undergoing the evolutionary stages of a certain final prod-
uct, we can obtain such a product by direct scientific synthesis. This is true
of an individual creation as it is of a style, a school, or of a whole culture.

These are the techniques embodied in the Theory of Regularity and Coordi-
nation, and these constitute the clue to the process of creation.

All non-uniform forms of regularity are the resultants of the interference of
two or more uniform simple periodic waves of different frequencies, brought into
synchronization. Such resultants, being the source of configurations, can be ob-
tained either by direct computation or by graphs. 4

When these resultants, the parent-shapes of all rhythms, are applied in

•See Appendix A.

♦See Chapter 3. '■Periodicity" of Part II. (Ed.)

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42

SCIENCE AND ESTHETICS

direct sequence to any of the primary selective series, they, in turn, produce
artistic scales, or sequent structures, which are the secondary selective series*^
These sets also vary in density, and, when they reach the point of saturation,
become identical with the primary selective series. Thus, we can state that the
secondary selective series is the result of rarefaction of a primary selective series .
which is the limit, and, thus, the dense set of the secondary series .

Secondary selective systems produce sequences known in mu sic as thg
rhythm of durations, of pitch, {i.e., pitclvscales) , of sequences of chord -progres-
sions, of intensities, of qualities, of attacks; in fact, of any or all components.
In linear design they are the source of sequences of linear, of plane or of solid
motion, resulting either in static configurations like spirals or polygons, or in
trajectories of various types, which include the participation of actual time in
the process of performance. More concretely, the resultants of interference
produce configurations from linear extensions, angles and arcs, which are the
equivalents of scales. Thus, we can refer to scales of linear extension, scales of
angles, scales of arcs, 5 etc.

The next stage of the Theory of Regularity deals with variations based on
general or circular permutations. Both general and circular permutations pro-
duce compensatory scales to the original scales, which may be called derivative
scales. The derivative scales together with their original, or parent-scale, con-
stitute one family of secondary selective systems. However, in practical appli-
cation, only such derivatives can be used as one family which result from permu-
tations of the same subcomponents. For instance, pitch-scales constitute one
family if the derivatives result from permutations either of the pitch-units or of
the intervals betwee n the latter, or of both such operations combined, but not
f rom jhe combinations of several such sets resulting from different operations.

\ThT originals and the derivatives, being brought into self-compensating
sequence, at the same time produce self-compensating simultaneity. In many
instances of artistic production, such simultaneity-continuity groups constitute
a complete composition"!]

After a family of scales has been developed into a compound secondary
selective system, new use may be made of the resultants of interference in the
form of coefficients of recurrence, or coefficient groups. Such coefficient groups
introduce a recurrence form into any original or derivative scale. The recurrence
of scale-units, applied to any original or derivative scale, transforms the second-
ary' selective scale into a master-pattern, or thematic motif, which in such a case
becomes the original operand and the tertiary selective scale, set, or system.

Further permutation of the master-pattern results in derivative master-
patterns, which, together with the original, produce a self-compensating simul-
taneity-continuity group.

The final compound set may constitute a complete composition or a complex
theme from which a complete composition may be evolved by means of further
permutation. This operation can be extended to any desired order and results
in any desired complexity in the final configuration, which generally retains its
homogeneous character.

s See Chapter 1. "Selective Systems" of Part III. (Ed.)

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The next basic technique, which follows the generation and the variation of
the secondary, tertiary and family-sets, is the composition of harmonic contrasts. 6
Harmonic contrasts, evolved as counterparts to original sets of whatever origin,
are based on distributive involution. Polynomials representing master-patterns
of any origin, type and complexity can be proportionately, i.e., harmonically,
coordinated with any number of other components, which may belong to the
same or to a different art form. Thus, not only two or more spatial configurations
can be proportionately coordinated, but a spatial kinetic configuration may be
coordinated even with a temporal configuration of sound, movement, lighting,
etc. This technique is of particular value in coordinating either the different
components of one art form, or the different art forms on the basis of harmonic
contrasts.

Involution-groups derive from the same source as the interference groups,
i.e., from the uniform sets. The selection of binomials or of polynomials from a
set is done according to the possible distributive form of the given set or series.
The determinants of the series are associated with the set of natural integers,
such as 4 series, ■§■ series, . . .-ft series. Some families are pure, i.e., they belong
to one series; and some are hybrid, i.e., they are the composite of several determi-
nants. Evidence shows that art works «f superior quality are "purer" than
those of inferior quality. This is true of traditional art as well as of individual
artists. The greatest works of art have been created, not by the innovators or
modernists, but by summarizers or synthesizers, who crystallized the experience
of their predecessors to the highest degree of perfection. Such perfection is
equivalent to consistency, and to correspondence between intentions and the
forms in which they are projected. In this sense, consistency means consistency
of the serial determinant, i,.e., strict adherence to a certain series: for instance,
the adherence of honeycomb cells to hexagonal symmetry, i.e., to series, of
starfish to-f- series, of the hereditary law of Mendel to-if-series, and of the forms
of growth to the first summation series, which consists of a group of determinants
like-£,-f."$ etc. These are symmetric in the Hellenic sense, i.e., they produce
an equality (constancy!) of ratios.

But the same forms of series participate in design, in sculpture, in poetry,
in dance, and in the other arts. In this sense, some forms of African and Asiatic
music, dance, and ornamental design belong to-§."§". and ■§■ series. Other forms of
European art belong to-f-, and series. The interference method, when ap-
plied to binomials of £ certain series, generates the entire evolutionary group
of that series. After reaching the point of saturation (binomials generate tri-
nomials, trinomials generate quintinomials and so on), the resultant of the last
order of that family is a dense set, i.e., uniformity of the limit.

These family- or style-series, after exhausting themselves in the last inter-
ference, either undergo the above-described involutionary harmonic develop-
ment, or hybridize themselves through intermarriage with other determinants.
There are many hexa-octagonal hybrids, for instance, in both music and design,
i.e., configurations which are combined products of$andf series.

The method of interference, as applied to evolutionary series, is the basic

•See Chapter 5. "Production of Combined Arts" of Part III. (Ed.)

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SCIENCE AND ESTHETICS

clue, possibly, to an understanding of the sciences and the arts. It is undoubtedly
the most powerful analytic tool known, being at the same time the source of
creation and of forecast. The latter is possible because once the tendency reveals
itself, the succeeding evolutionary stages can be determined a priori.

Once all these resources of regularity and coordination are disclosed, the
most ambitious undertakings in the field of art creation can be easily accom-
plished, as the series and the involution technique take care of both the factorial
and the fractional sides. This means that form, growing as a whole, is at the
same time coordinated with its own individual units and their configurations.
Thus, the whole evolves itself on the basis of compensation and/or contrast.

C. Other Techniques of Variation and Composition

Further techniques evolve from other sets than those already described.
Among these, the most prominent are the arithmetic progressions, geometric
progressions, summation series, involution -series and various other types of
progressive series, all of which may be used as the secondary selective systems,
i.e., configuration scales. 7 Such sets and series control the effects of variable
velocities and of growth. The effects of positive and negative acceleration are
the domain in which such scales provide the basic material of structure and coordi-
nation. Coordination is based on variability of phases, coefficients and direction
(positive and negative acceleration). These scales, being coordinated with their
own converses, produce resultants of interference of their own kind. These re-
sultants, in turn, become scales, master-patterns, etc.

Another important phase of composition, pertaining to variability and ex-
tension, is quadrant rotation. 8 Optical and acoustical images can be subjected
to this form of variation, which results in four equivalid variants of the original
(including the original). The four variants can be further extended by means
of the permutation technique, which, in turn, can be extended to any desired
order. These variations do not change the inherent relations of the original
configuration, but merely project it into a new geometrical position. Quadrant
rotation is a technique which is both basic and natural.

Still another basic technique of variation and composition is coordinate
expansion. 9 It may be performed either to the abscissa, thereby affecting the
general component, or the ordinate expressing some special component. Coordi-
nate expansion may be positive or negative (contraction); it may also assume vari-
ous forms, such as arithmetic, geometric, logarithmic expansion or contraction,
etc.

Different forms of coordination of the two axes of expansion , corresponding
to the two coordinates, result in a variety of types of geometrical and optical
projection. Control of proportions in homogeneous and heterogeneous space
can be accomplished by this technique. As a resource of composition it is rather
limited in the field of acoustical images, largely due to instrumental and percep-
tive limitations, but has an unlimited scope in the visual arts.

'See Chapter 2. "Continuity" of Part II. (Ed.)

•See Chapter 10. "Quadrant Rotation" of Part II. (Ed.)

*See Chapter 11. "Coordinate Expansion" of Part II. (Ed.)

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In music, being applied to pitch-coordinate, geometrical expansion trans-
forms music of one style of intonation into another, modernizing the original
when used in its positive form. For example, expansions of Bach or Handel
produce Debussy or Hindemith. Geometrical expansion of pitch affects the
primary selective systems in such a way that their original points of symmetry
become their own squares and cubes in the positive expansion , and their square
or cube roots in the negative expansion. Thus, for instance, the contraction of a
tuning system from the yfl to the fyl would permit the performance of semitonal
music in the quarter-tone system. Positive expansions may be performed in the
original system in which they produce configurations based on the rarefied set.
In the visual arts, this technique produces forms of optical projection and aber-
ration such as those for which El Greco and other artists of unusual perspectives
are famous.

As sets and configurations grow through their own variability, resulting in
different forms of saturation, the next important technique is tha,t which pertains
to density. 10 A density group must be considered a configuration of density or
saturation. A dense group of density is the limit-group of saturation under
specified conditions. A dense audible image requires the use of all parts of the
score in a dense distribution of pitches. A dense visual image is saturated with
configurations, which may overlap, and which are confined to the visual area,
such as the canvas or the screen . In music, a monody is not a dense set, compared
to harmonically accompanied polyphony. In painting, one flower occupying
only a small portion of the total area is not a dense set, compared with a battle -
scene completely crowding the picture.

The technique of composition of density-groups, their variation- and coordi-
nation, controls the degree of saturation and its distribution throughout the
whole. The variants of the original density configuration, i.e., the master-
pattern of density, in turn produce simultaneity-continuity density groups.
The same configuration, of whatever origin, may coexist with its own variants
of density, which often implies an evolutionary stage. Density groups, together
with their variants as configurations, are subject to positional rotation. Such
rotation may follow both coordinates and intercomposes their phases. This
latter stage represents the ultimate degree of refinement in composition.

Homogeneity of character in the work of art defines the necessary quantity
of master-patterns pertaining to various components. Such master-patterns
constitute the thematic material of the composition, and may be coordinated
into a whole by any of the basic techniques described above , or any combination
of the latter.

One cannot fail in evolving a work of art according to this method of specifi-
cation, selection and coordination. The specifications must be chosen in accord-
ance with semantic requirements, which define the purpose of the production.

I0 See Chapter 12. "Composition of Density*' of Part II . fErt.)

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D. Pragmatic Validity of Theory of Regularity

Seven points upon which the pragmatic validity of this theory rests:

(1) It establishes esthetic principles which remain true in any special in-
stance.

(2) It provides a foundation for more efficient creation and a more objective
criticism.

(3) It does not circumscribe the freedom of an individual artist, but merely
releases him from vagueness by helping him to analyze and to realize
his own creative tendencies. It gives him a universal knowledge of his
material: the principles and the techniques of this system permit an
infinite number of solutions, which satisfy any requirements set forth
by art problems.

(4) It offers the student of this theory a manifold of techniques that enable
him to handle individual and combined art-forms. Since these tech-
niques are interchangeable and inter- related, designs and melodies|inay
be plotted- as graphs, and the student may dance or sing a design, or
translate a melody into a drawing.

(5) It is scientifically valid in that it establishes the basic principles under-
lying creative processes and correlates esthetic reactions with generating
excitors (i.e., the works of art) of definite forms and defined variations.

(6) It stimulates art production and reduces the years of training, thereby
making creation a process associated with pleasure, accomplishment and
satisfaction .

(7) Its social significance lies in the fact that it leads toward unity and
tolerance and away from disunity and intolerance through an under-
standing of the basic principles of inter-relatedness and functional
interdependence .

This theory has been presented in part before various learned societies, in-
cluding American Institute for the Study of Advanced Education, Mathematics
Division of the American Institute of the City of New York, the Mathematicians
Faculty Club of Columbia University, and the American Musicological Society.
It has also been offered by the author in the form of courses and lectures at
Teachers College of Columbia University (Departments of Mathematics, Fine
Arts and Music), at New York University, and at the New School for Social
Research .

Students of this theory included educators, architects, artists, designers,
composers, and conductors. Some of them were celebrated artists when they
came to study with me, and some attained prominence, partly as a result of their
studies.

Some of the products of this theory, including my own compositions and
the compositions of my students, have been presented in various forms in the
field of symphonic, chamber and applied music (motion pictures, radio); in art
exhibits, such as at the Architectural League of New York; and in science ex-
hibits, such as that at the Mathematics Museum of Teachers College, Columbia
University.

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Most contemporary minds occupied with the thought of employing engineer-
ing to serve art, have restricted their research to telecasting, television, enlarging
visible images, amplifying sound, expanding the range of visibility and audibility,
etc. Mathematical analysis of the process of composition and production in art
media and by means of electro-mechanical (acoustical, -optical, etc.) synthesis
has been for years the subject of my research. The theory formulated in this
work is an objective system in the sense in which a system of algebra or geometry
is objective. Through the application of principles and techniques evolved in
this theory, works of art can be produced by computation and plotting, and
therefore can also be realized mechanically in an art medium, i.e., through
automatic composition and performance.

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THEORY OF REGULARITY AND COORDINATION

Each system is valid when functioning within its own
strictly defined limits and its own operational conditions
(laws) .

We discover in the evolution of method that new processes
call for new operational concepts, new terminology and
new symbols.

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SO THEORY OF REGULARITY AND COORDINATION

Consecutive Selective Techniques

(1) Notation:

a. universal system of mathematical notation for ail art forms, structures
and processes:

b. graphs.

(2) Classification of Formulae:

a. series (selective)

b. distributive and combinatory

c. definitive (identities)

d. quantitative, computative (equations)

e. combined: computative-distributive (involution)

(3) Selective Series:

a. primary

b. secondary

Components and configurations

(structures, assemblages, forms)
Continuity series (secondary selective series):

closed and progressive forms of symmetry.

(4) Derivative Harmonic Groups: the resultants of interference and their in-
volutionary forms. Use of these groups as coefficient recurrence-groups.
Evolutionary series (interference development).

(5) Synchronization (coordination): primary, secondary, etc.

(6) Kinetic Geometry (space theory) [proportions, kinetic forms and images,
diffusion forms, logarithmic variations of coordinates, quadrant rotation,
cylindric rotation through abscissa or/and ordinate, projections: rectangular,
spheric, etc.]

(7) Physics: color, space, motion, frequency, intensity.

Applications of Mathematical Techniques

(1) Art Forms (definition and classification).

(2) Art Components (primary and secondary scales of components and their
interrelation); (interrelation of components in an individual art form).

(3) Art Processes (the techniques of production).

(4) Correlation of Art Forms.

(5) Samples of Synthetic Art in the Individual and Combined Forms.

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CONTINUUM

A. Definition of an Art Product by the Method of Series

' I 'HE mental growth of humanity, as revealed in scientific thinking, may be
stated as a tendency to fuse seemingly different categories into a complex
unity, into which previous concepts enter as component parts. The evolution of
thought is a process of synthesizing concepts.

Assuming this as a methodological premise, we can build complex concepts
from concepts which previously seemed dissociated. Such concepts enter into
a series as definitely located terms. Thus, a "record," an "impression," or a
"stimulus" may be regarded as only one of a series of terms with respect to its
place within different series.

The evolution of the concept of a spectrum provides an analogous case.
Sanscrit literature shows that the three primary colors linguistically and other-
wise evolved into the whole spectrum, and that this evolution may be determined
through serial development from the primary colors.

Example: (1)

A simplified interpretation of color using standard terminology will easily
show how the method of evolving series can be applied. Selecting from the infi-
nite spectrum of hues a limit within a certain primary yellow and primary blue,
we may observe the following growth of the series.

Y B

Y G B

Y YG G BG B (additive system)

[We find the same happening in musical pitch: a scale grows through inserting
definite tones between tonal limits, whatever they may be: (?■."§. f") •

Example: (2)

2 3_

3 2

1 2. 1

3 2 2

11111.

3 3 2 2 2

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THEORY OF REGULARITY AND COORDINATION

In a scale of increasing frequencies and applying the 4 (octave) adjustment, the
following development may be seen:

3_ _8_ _2_ ' _9_ 4_

4 9 2 8 3

(The note equivalents of these ratios are: G, Bb, C, D, F)

As in this case, all the tones represented through definite ratios of frequency
may be regarded simply as different terms of a series with respect to their places
within the series^]

It is not sufficient to examine a work of art in order to produce the phenome-
non of an "art product." A number of conditions must be fulfilled in order to
obtain an art product. A work of art is primarily an idea expressed through the
means of art — its material. All art material has definite properties, i.e., it is a
complex of sound, a complex of light, a complex of mass, etc. Each art material
consists of a number of components, such as pitch, intensity, quality of sound;
hue, intensity, saturation of color; etc. 1

An idea, therefore, can be expressed through correlation of the components
of art material. When this condition is realized, we have the first requisite for
the building of an art product.

Thus the first term is: — an idea as realized through the correlated components
of an art material.

The value of such an idea is to produce a stimulus, sensory, motor, or mental.
A sensory stimulus may or may not transform into a motor or mental form. A
motor stimulus occurs, for example, when Charleston music stimulates muscular
movements which result in an appropriate dance form, or when military music
stimulates one human being to attack another. An illustration of a mental
stimulus would be Sposalizio by Franz Liszt, which was stimulated by the
Sposalizio of Raphael. The ancient Greeks did not believe that anything was
achieved if a musical work did not produce mental stimulation. 2 Thus, stimulus
is a term in a chain that results in the phenomenon of an art product. As
soon as we have obtained these two terms ("idea" and "stimulus"), we can de-
velop the rest of this complex concept by the method of series. "Idea" and
"stimulus" are the two extreme terms of the series:

IDEA STIMULUS

There must be a medium between an idea and a stimulus — a purely physical
medium like air or an electromagnetic field. Thus, we have a three term series:

IDEA MEDIUM STIMULUS

In order to exist in the medium, an idea should be generated physically.
This gives us a generator. A "generator" in music can be an acoustical instru-
ment; in the art of photography, an optical instrument. The "generator" will
produce actual physical oscillations. To analyze these physical oscillations, a
transformer is necessary. The human brain is such a "transformer."

'Art material is treated analytically as a system; 'Plato. Polyteia.
its components, as parameters forming such a
system.

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The next stage in the development of this series is:
IDEA generator MEDIUM transformer STIMULUS

An idea can be expressed physically on a record. Such a "record" in music
can be the sound track on a film, a record groove, a musical score in musical
notation, a musical score in graphs, etc.

A "generator" requires a transmitter which will propagate the oscillations
through a medium in a form adequate to the record. In music a human performei.
usually is the "transmitter." A mechanism may be substituted for the per-
former. 5 A "transformer" obtains its material supply from the receiver, which
may be an organ of sensation such as the eye or ear. Finally, a stimulus can be
obtained from a transformer only through the impression (made physiologically
and electrochemically upon our brain cells, or psychologically by affecting our
mood).

This completes the nine term series representing the complex concept of an
Art Product:

IDEA record generator transmitter MEDIUM

receiver transformer impression STIMULUS.

There can be two types of results in the development of an art product:

1. If the stimulus is sensory when the series is completed, the experience
will fade out. This aspect of the series will be graphically a straight
segment.

2. If the stimulus is -mental or motor, the final term-stimulus will trans-
form into a new idea (I 2 ) or into action. Then the evolution of the
series will go through the same process, starting from the new point.
This aspect of the series will be graphically an infinite cylinder built
through consecutive coils,

/,

7 * 1

Figure I. Chain relationship of idea and art product.

where "I" will represent a new idea resulting from consuming and assimilating
an art product.

'In an orchestra, a conductor is the transmitter transmitter at the same time, while the piano is a
and the orchestra is a generator. In the player- generator. The generator is a passive mechan-
piano, a perforated paper roll is a record and a ism; the transmitter is an active mechanism.

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THEORY OF REGULARITY AND COORDINATION

This chain process is logically similar to the continuation of a race through
the reproduction of an individual. This is the only way to preserve the products
of evolution. In some cases, for example in painting as compared to music, cer-
tain terms in the series will be missing because the record (canvas), the generator
(canvas) and the transmitter (canvas) are identical. In the case of a compound
art form (like sound cinema), we shall have parallel simultaneous series, abso-
lutely alike or different.

An art idea can be transmitted from one individual to another if the sum
total of their previous experiences in art is not too different. In order to produce
an idea at the end of the series, the perceiving individual must be equipped with
discriminating experience. This allows him to determine (or at least to have a
general orientation in) the relations and correlations of parametral values. He
must adapt himself to the absolute values of extensions, directions, positions and
durations, in order to enjoy all the variations and modifications of an idea. He
must be well-equipped particularly with discriminating experience in the field
of special components.

Imagine an individual whose sense of pitch discrimination is such that he
cannot recognize an increase or decrease in frequency of sound within ratio. 4
A melody by Chopin will be entirely distorted in his audible apprehension.
Imagine a color-blind individual looking at a painting. The color relations will
be beyond his visual apprehension. Finally, in order successfully to build an art
product, concentrated and focused attention is necessary; otherwise, many ele-
ments will escape and the whole may be destroyed. In other words, the active
cooperation of a consuming individual is absolutely necessary.

For the successful projection of an art idea:

1. all the terms of the series must be valuable or reliable in the adequacy
of their functioning (the idea should have value, the record must express
adequately the idea; the generator should be plastic, precise and reli-
able, etc.).

2. the consuming individual must be reliable in his focused attention and
parametral discrimination. 5

A change occurring in any term of a series might have an effect on some other
terms of the same series.

If in a series forming an art product we take Rd (record), Rr (receiver) and
In (impression):

Rd! — Rri — Ini

*One whole tone.

'There i3 always a certain amount of the un-
known in a work of art which we cannot discrimi-
nate until we go through several analogous ex-
periences. If the amount of the unknown equals
zero, we have perfect banality. If it is not too

great, the majority enjoys it immensely. If it is
too great no one enjoys it at first, for the simple
reason that he does not understand the language
in which it is brought to his attention. The ideal
for a discriminating group of people lies between
these extremes.

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55

and if we change the Rr with the desire to preserve the same impression, we are
forced to change the record:

Otherwise, the result will be:

Rd,— Rr, — In,

Rdi — Rr, — In,

In other words, if our perceptive apparatus changes, we cannot apprehend
a work of art in the same way any longer. An authentic performance of Bach's
Fugues on a harpsichord cannot impress us in the same manner and to the same
degree that it impressed Bach's contemporaries. Here is an answer as to whether
a transcription or an arrangement of old music is legitimate. It is not only
legitimate but necessary if we do not wish to miss the enjoyment our ancestors
experienced.

DEVELOPMENT
OF THE COMPLEX CONCEPT
OF AN "ART PRODUCT"
BY THE METHOD OF NORMAL SERIES

b. . . .y

b m y

b h m s y

b e h j m. . . .p s v y

Idea. . Stimulus
Idea. .Medium. .Stimulus
Idea. .Generator. .Medium. .Transformer. .Stimulus

Idea. . Stimulus.
Record.. Impression..
Generator. . Transformer. .

Transmitter. . Receiver. .

Medium . .

Figure 2. Development of an art product by the method of normal series.

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B. Parametral Interpretation of a System

In the first edition of his General and Special Theory of Relativity, Albert
Einstein based his physical interpretation of the universe on Minkovsky's and
Riemann's geometrical premises. Thus, he established for all space-time mo-
menta, including their electro-magnetic behaviour, a hypothetical, geometrical
system of parameters (measuring lines): Xi, Xj, X 8 , X«. It is neither essential
for these parameters to have actual existence in this universe nor for them to
correspond to our sensory perception. They are assumed in a purely mathe-
matical (logical) sense as tools that enable us to explain what we know as the
physical universe. 6

There is no reason why analytical study of the world of art should methodo-
logically differ from that of the physical universe. Our definition of an art prod-
uct opens the door to such an interpretation.

What is art? We sit in a concert hall and listen to a piano recital. What
stimulates us emotionally and mentally? The pianist touches the keys, which
excite the strings; the strings move in transverse vibrations, which are sent
through the air medium in longitudinal waves; the longitudinal waves affect our
organ of hearing. This purely physical complex transforms into a physiological,
i.e., electro-chemical one, and, finally, into one of tone. The pianist learns
which keys to strike, and in what progression and manner, from a highly sym-
bolic and imperfect record called a musical score. He tries to present an ade-
quate projection of this symbolic record, which contains the composer's idea as
expressed in simultaneous and consecutive tone relations. Is this all? We be-
lieve it is, insofar as music is concerned, providing that the method of recording
the composer's ideas is adequate and reliable.

One may ask: But what about the personality of the performer? his "mag-
netic" power? Then we may also ask: What about the personality of the alluring
companion whom you brought to the recital and whose influence upon the re-
sultant of your mood during the recital should be rated at 35% at least? All
these factors must be taken into account since this is a problem of psychological
resultants produced by the entire complex of stimuli acting upon us at the time.
But these are not aspects of the art of music as such .

We can learn a great deal about moods stimulated by music if we learn
enough about music itself. And this knowledge can be gained only from a reli-
able record. From a purely acoustical viewpoint, music can be analyzed from
phonograph records, film sound tracks, oscillograms, etc. But this would pro-
vide little help in understanding a musical idea, and for the same reason that a
microscope would be of little use in determining the contour of a continent.
An oscillogram gives a splendid projection of a sound wave at a given moment,
but with such detail and complexity that all the points of musical interest are

•During the year 1933 Einstein adopted the five physical momenta, thus requiring the introduc-
parameter system (Xi, Xj, Xj. X«, Xj) because tion of a new parameter,
the previous system failed to explain certain

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too far removed from one another to mean anything musically. In both cases,
the method is unsuited to the dimensions. From the physical point of view,
it makes no difference whether chords change at one speed or another, while
musically such variations in speed might completely transform the artistic mean-
ing.

Why should one chord be followed by another and why should changes in
speed be necessary? This cannot be answered by the physics of sound. The
fact that the science of musical sound (not the science of musical composition),
during the 5000 years of its existence, did not explain the mechanism of musical
composition is sufficient evidence that acoustics is not adequate to provide such
an explanation .

Herman von Helmholtz in the introductory chapter to his Sensations of Tone
("Plan of the Work") writes: "Questions relating to the equilibrium of the sepa-
rate parts of a musical composition, to their development from one another,
and their connections as one clearly intelligible whole, bear a close analogy to
similar questions in architecture." Friedrich Schlegel said: "Gothic architecture
is frozen music." By inverting this proposition we can say: music is fluent (or
animated) architecture. But if gravitational tones are today to be explained
geometrically, we can ascertain that music is a fluent geometry, i.e., geometry
using the time parameter. With a sufficient number of parameters, we can
explain all the "laws" pertaining to a functioning system, whether it is a system
of art or of the whole physical universe. When one ascertains that harmony
or counterpoint can be explained, but not melody, it merely indicates that the
necessary musical parameters have not yet been found, and that the interpreta-
tion provided for harmony and counterpoint may be correct only in certain
cases.

The subtle points in music technically correspond to gravitational or electro-
magnetic phenomena (accumulation and discharge, attraction and repulsion),
and once they acquire the proper geometric interpretation, all the "mysteries"
of dramatic quality in music are revealed. This analysis can be made by means
of the adequate record of an art idea. A comparative study of such records leads
to the establishment of general laws of composition in art. 7

A continuum is a system of unlimited parameters. In terms of measure-
ment, parameters are the extensions. Any individual art form (music, sculpture,
etc.) is a continuum, i.e., a system of parameters representing the art compo-
nents. Thus, we can speak of different art forms as different continua (continuum
of sound-music; continuum of mass-sculpture, etc.) Every individual art con-
tinuum consists of two kinds of components:

1. general components (time, space);

2. special components (frequency, intensity, quality).

7 The function of an interpreter of music, who
is the "generator" in the whole complex ol a
"musical product," approaches zero when the
"record" includes all the specifications of a
"musical idea" and when his intentions are sin-

cere; in other words, when he is trying to present
an adequate projection of a "record." Thus, the
elimination of a living performer is a natural re-
sult of the process of evolution. Interpretation
will probably survive as a hobby.

Digitized byLiOOglC

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58

THEORY OF REGULARITY AND COORDINATION

An individual art continuum is related to a sensory continuum; the param-
eters of an art continuum correspond to the functioning of the organs of dis-
crimination. If sound, as a musical continuum, consists of parameters of fre-
quency (pitch), intensity (volume), quality (timbre and character), there must
be corresponding discriminating units in our organ of hearing. These discrimi-
nating units produce corresponding reactions. 8 Thus, the cochlea in the inner
ear, for instance, respond at definite rates of frequency with their microscopic
stringlike units (Cortis arches) of varied length. The degree of air pressure of
a sound wave, resulting from different amplitudes, affects the ear-drum likewise.
Quality is a complex component resulting from frequency, intensity and wave-
phases in their different relations. Discrimination of the quality of sound does
not require new organs of discrimination.

The general categories of our mind, according to Kant, provide us with an
orientation in space-time relations (general components) , while our organs of
sensation enable us to discriminate our sensory perceptions (special components).

Different individuals have differently developed organs of sensation. The
discriminatory functioning of these organs can be developed greatly through
training. These human abilities have usually been underestimated. The capacity
in pitch discrimination of an average house dog, according to Pavlov's* experi-
mentation in conditioned reflexes, is up to \/2, i.e., one fortieth of an equally
tempered whole tone 10 in the twelve step equal temperament. A musically in-
clined individual, with fairly good pitch discrimination, notices the difference in
about one one-hundredth of an equally tempered whole tone ( V^)-" It does
seem strange that some musicians think a quarter-tone too small an interval
and too hard to discriminate.

C. The First Group of Art Forms (One System of Special Parameters)

We shall now discuss the different parameters of various art continua.
We shall describe the individual art forms (art continua) that are possible, and
the complex art forms (systems of continua) that can be deduced from them.
Classifying art forms through the number of their general parameters, we obtain
the following scheme:

The first group requiring one organ of sensation at a time and one system of
special parameters .

•According to recent studies, these reactions are
electro-chemical processes.

'Ivan Pavlov: Conditioned Reflexes.

I0 A whole tone in an equally tempered twelve-

unit temperament

Digitized byGoOgle

"See Prof. Carl Seashore's materials on tests
in pitch discrimination. I also had the opportun-
ity of making tests by means ot an electric organ
constructed by Leon Theremin with variations in
pitch up to jfe of a whole tone in the twelve-unit
equal temperament.

Original from
UNIVERSITY OF MICHIGAN

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by^jUUgl^ UNIVERSITY OF MICHIGAN

60

THEORY OF REGULARITY AND COORDINATION

Figure 4. Diagram of art forms.

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

CONTINUUM

61

KINETIC ARTS (musical)

One general parameter — time.

Sensation

General Parameter

System (a complex) of special parameters

Hearing

Time

Sound (1)

Touch

Time

Mass 1 * (2)

Smell

Time

Odor (3)

Taste

Time

Flavor (4)

Figure 5. Kinetic arts with one general parameter.

STATIC ARTS (plastic)

Two general parameters — two dimensional space — area.

Sensation

General Parameters

System (a complex) of special
parameters

Sight
Sight
Sight

Coordinates Xi, X2
Coordinates Xi, Xi
Coordinates Xi, X2

Light 13 (5)
Pigment (6)
Surface" (7)

Figure 6. Static arts with two general parameters.

"(2) Marinetti proposed touch as an art torm
in his manifesto on Tactilism in 1920.

"(5) Light as a source of illumination of a trans-
lucent (glass) or a transparent (dampened cloth)
surface.

Digitized by GoOgle

"(7) Different visible textures of materials used
as the material of art. Solid (wood. wire, glass,
cork, rubber, etc.) and liquid; also photography
and patterns used as elements of texture.

Original from
UNIVERSITY OF MICHIGAN

62

THEORY OF REGULARITY AND COORDINATION

KINETIC ARTS (musical)

Three general parameters = time X 4 (1) + area Xi, X 2 (2).

Sensation

General Parameters

System (a complex) of special

parameters

Sight

X„X ? , X«"

Light 16 ( (8)

Pigment 17 (9)

Surface 18 (10)

Figure 7 . Kinetic arts with three general parameters.

STATIC ARTS (plastic)

Three general parameters = volume (a system of planes).

Sensation

General Parameters

Special Parameters

Sight

Xi, Xi, Xj

Light (11)

Pigment (12)

Surface (13)

- Mass 19 (14)

Figure 8. Static arts with three general parameters.

'*x 4 = t.

"(8) Light source as such = mobile light.

,7 (9) Pigmented mobile liquid; pigmented im-
ages on a plane transforming in time, chemically,
mechanically, or optically. A rchipentura (or in-
stance.

Digitized by GoOgle

"(10) Mobile textures of materials: transfor-
mations obtained chemically, mechanically or
optically; geometrical transformations on a plane
with or without perspective.

"(14) Solid mass; planimetric clusters.

Original from
UNIVERSITY OF MICHIGAN

CONTINUUM

63

KINETIC ARTS (musical)

Four general parameters = volume (Xi, Xi, X 3 ) + time (X«).

Sensation

General Parameters

Special Parameters

Sight

Xi, X21 Xs. X4

Light"
Pigment 3
Surface 12
Mass 13

(15)
(16)
(17)
(18)

21

Figure 9. Kinetic arts with four general parameters.

Considering the fact that our perception of the outer world is based on two
categories of consciousness, space and time (corresponding to the four mathe-
matical parameters, Xj, X 2 , X 3 , X 4 ), we may classify all forms of art into two
groups — static and kinetic.

To the first group belong those arts defined as forms crystallized in space,
which can be perceived by sight only. They do not change in time after the
process of composition is completed. They can be observed from various sta-
tionary positions, each point of observation revealing a different form.

To the second group belong those arts that can be defined as a process of
generation, transformation and degeneration of forms in time and continuity.
They change in time after the process of composition is complete. This process
can be perceived by any of the organs of sensation: sight, hearing, touch, smell
or taste. All the art forms appealing to the different organs of sensation involve
changeability in time, while sight may or may not require such a condition.
Visual perception can be directed towards changeable as well as unchangeable
forms. A kinetic impression may be obtained from static optical forms by a gradual
displacement of the object, or of the point of observation (the position of the
observer) .

In art forms pertaining to touch, smell and taste, changeability in time can
be produced in two ways: the excitor {generator) moves while the perceiving

w (15) Mobile light. cally, mechanically or optically .

*'(16) Pigmented mobile solids (solid masses; w (18) Solid (solid masses; planimetric clusters)

planimetric clusters), liquids, gases. hard, soft, gummy, liquid, gaseous — transforming

B (17) Textures transforming in time chemi- geometrically. «

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64

THEORY OF REGULARITY AND COORDINATION

individual (smeller, toucher, taster) is in a stationary position; or the perceiving
individual moves in space while the excitor remains stationary. For the hearer,
only one form of apprehension is possible. He is stationary and the sound evolves
before him in time.* 4

By eliminating our organs of sensation, an impression may approach zero.
By excluding an organ of sensation that is not engaged, we concentrate better
on the excitor appealing to other organs of sensation. Thus, many people close
their eyes while listening to music. *

Any system of special parameters (components of an individual art material)
can be reduced to three parameters:

1 . Frequency (or form)

2. Intensity (dimension or size)

3. Quality (texture or character)

Frequency and intensity are the conditioning parameters while quality is itself
conditioned by the correlation of frequency and intensity, and is therefore a
derivative product of both. 26

A complete classification of all possible art continua through their general
and special components (parameters) can now be achieved. We shall rely in
part on the previous classification.

Art Form Number One (sense of hearing) has two aspects:

(a) art of audible sound (music) 26 , with a system of special parameters —
sound = frequency (pitch), intensity (volume = loudness), quality
(timbre and character of sound) ;

(b) art of the audible word (poetry) 27 , as an independent declamation or as
- part of a play with a system of special parameters — word (sound plus

its semantic connotations) = frequency (pitch and inharmonic com-
plexes), intensity (loudness), quality (1. sonorous character of the
, material; 2. sonorous character of the reciting voice), and the param-
eter of a plot built through a system of correlated poetic images as
units. 28

"The movement of a hearer or of the source of
sound production varies pitch to an extent not to
be taken into account in music, as we conceive it
today.

"See D. C. Miller: The Science of Musical Sounds
(1926).

"Musjc of sound (art of audible sound).

Digitized byGoOgle

I7 Music of verbal images and sounding words
(art of the audible word suggesting the relation of
images) .

18 Many attempts have been made to establish
poetry as a form free from the literary connotations
of words (incantation) .

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CONTINUUM 65

Art Form Number Two (sense of touch):

Art of touchable mass with a system of special parameters — mass (in all its
perceptible aspects) =

(a) quality of mass — texture 29 , conductibility:

1 . due to the atomic structure.

2. due to the speed of motion (movement of the mass).

(b) temperature:

1 . due to general conditions.

2. due to the speed of motion.

(c) intensity:

1 . due to weight or pressure.

2. due to electric conductibility (discharges).

Though this art form does not exist independently, we exercise our discrimi-
nation of tactile perception in every-day experience 30 : vibratory massage, sexual
experience, the fondling of animals, a facial in a barber shop, diving, sailing,
selecting fabrics, etc. The art of touch, now hypothetic, undoubtedly has a
chance to play an important part in our lives in the near future when certain
social standards are revised. 31

Art Form Number Three (sense of smell):

Art of smellable odor with a system of special parameters— odor =

(a) quality (character of odor).

(b) intensity (of odor).

(c) density (quantity of odor in a given space) .

(d) temperature (as perceived by the nose).

(e) humidity (of the air).

This is a hypothetic art form found in every-day experience. 32 Some appli-
cations have been made in commercial advertising in America. 33

"Hard. gummy, soft, liquid.

"Look at advertisements: "a lovely skin in-
vites romance," "the skin you love to touch,"
"that well-groomed look" (visual texture, but
suggesting tactile impression [complex sensation)).

"There is an interesting problem in the art of
touch; that of tactile, quantitative illusions ob-
tained through displacement of organs of touch,
fingers, doubling or tripling an object.

"Its application is as old as rrligion. Incense
has been used in religious ceremonies and also in
homes. Woolworth sells it!

"Perfumed paper, tobacconized paper, etc.
"It is not the way you look, it is not the way
you talk, it is the way you smrll" (from a perfume
advertisement) .

Digitized byGoOgle

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THEORY OF REGULARITY AND COORDINATION

Art Form Number Four (sense of taste):

Art of tastable flavor with a system of special parameters — flavor:

(a) quality (character of flavor).

(b) intensity (strong or weak flavor).

(c) density (the degree of concentration).

(d) temperature (of the material).

(e) quantity (of the material taken at a time).

(f) quality of resistance (texture of the flavor: liquid, soft, gummy, hard).

In order to experience works of art based on the sensation of taste, it is
enough to taste or to chew without swallowing. This makes an essential differ-
ence between the art of taste and gastronomy. Such an art is potential in chew-
ing (tobacco, betel, gum). Professional liquor tasters usually do not drink
alcoholic beverages.

Temperature is a common component of the last three forms of art (Nos.
two, three and four), which are at present hypothetical.

Art Form Number Five (sense of sight):

A rt of visible light** with a system of special parameters — luminescent paint** =

(a) frequency (hue).

(b) intensity (luminosity) .

(c) quality (saturation).

This art form became a favorite several decades ago when it was used as a
part of the setting of vaudeville shows. The luminescent effect is produced from
especially prepared paints, illuminated by ultra-violet rays that make the paints
visible in the dark. It might be considered as "luminescent painting" visible
only in the dark. The color system is additive. This art form requires a light
source in front of the painted surface.

Art Form Number Six (sense of sight) :

Art of visible pigment with a system of special parameters — pigment 5 ' =

(a) frequency (hue).

(b) intensity (luminosity).

(c) quality (saturation) - .

"Light as a system of parameters has been used "Non-luminescent material (1) oil, water color,
and may be used in luminous screens (luminescent crayon, etc.. in art done by hand with brush or
murals) with the source of illumination behind the without; (2) chemical pigmentation done by photo-
screen, graphic chemicalia.

"The source of illumination, ultra-violet rays,
is an invariant and cannot be considered a param-
eter.

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CONTINUUM

67

So far, two forms of this art exist:

1 . Where the light source is in front of the surface (opaque surface) paint-
ing, etching, fresco, photography 37 ; the color system is additive.

2. Where the light source is on both sides of the surface: (translucent sur-
face) painting on glass as in lantern slides, curtains, etc.; the color sys-
tem is subtractive.* 8

In this case the source of illumination (white light) is an invariant and as
such cannot be considered a para meter

The essential difference between forms five and six is that while luminescent
paint gives the effect of a light source, pigment, even though used on translucent
material, does not.

Art Form Number Seven (sense of sight) :

Art of the texture of visible surface with a system of parameters — texture:

(a) frequency of recurring elements and their dimensions 40 (density of
structure) .

(b) intensity 41 (extension into the third dimension).

(c) quality 42 (molecular structure of matter, absorbing-reflecting reaction
on light).

This art form was presented as a definite movement at the end of the first
decade of the twentieth century by the "futurists." It has failed, probably be-
cause of poor ideas and poor realization. Fabrics, fragments of photographs,
newspapers, household tools and what-not were used as materials. Worthwhile
attempts were later made to apply it to window displays, commercial advertising
and interior decorating. Some successful experiments were made in book covers,
using different types, and photomontage as elements of texture. 45

Strictly speaking, the difference between design as an art material and tex-
ture of visible surface is purely quantitative: microscopic configurations form
texture.

There are many hybrid forms where the effects of visible color and the tex-
ture of visible surface are combined: "pointillism" in painting; illumination of
Niagara Falls by powerful light sources, where falling water forms a most im-
pressive texture and provides a unique screen offering unusual opportunity for
luminous color effects.

"In this classification an optico-chcmical pro- "Patterns of design as elements of texture,

cess substitutes for pigment and is realized through "The bas-relief quality, producing light-shade

a mechanical device (camera). effects.

M A static color composition projected on a ^Solid (from "hard" to "soft"), liquid, gaseous,

screen is also in this group. 4, A honeycomb is an illustration of homogene-

**When the source of light is variable, such an ous texture with small but distinctly noticeable

art becomes kinetic. elements (patterns).

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68

THEORY OF REGULARITY AND COORDINATION

Art Form Number Eight (sense of sight) :

Kinetic art of visible light projected on a plane surface. (Art form number five
in its kinetic state, functioning in time.)

There are two distinctly different forms of the kinetic art of light. The
first deals with the light source as such (Rimington, Wilfred, Klein) and is known
as the art of color-music, producing a visible reality.

The second uses the light source as a medium for projecting successive photo-
graphic records representing consecutive phases of a moving actuality. It re-
produces an actuality formerly existing as an event. This event is first analyti-
cally photographed and then synthetically reproduced in motion. It is the art of
the cinema, a typical 20th-century product. It has a number of possibilities:

(a) It can record and reproduce an actual event (action and acting).

(b) It can record an actual event and reproduce it at an entirely new speed
(much slower or much faster) , 44

(c) It can produce an effect of actuality by recording a number of drawings
that represent consecutive phases of motion. This has been developed
in "trailers," "cartoons" and "kinetic abstractions," and is based on
"frame by frame" (single shot) technique.

(d) Optical distortions and multiplications of a recorded actuality (system
of mirrors) .

(e) The multidimensional effects obtained by montage intercomposition of
continuity and of frame: interpenetrating translucent actualities.
This method is an essential supplement to the three-dimensional effect
(depth), obtained through motion on a two-dimensional surface (screen).

All these possibilities provide a splendid opportunity for the development
of an independent art of kinetic images, so-called "abstract cinema," until proper
optical instruments using light source as such (not a record, not a film) are de-
vised . 46

Art Form Number Nine (sense of sight) :

Kinetic art of visible pigment transforming on a moving surface. (Art form
number six functioning in time).

This art form is based on the movement of pigmented surfaces consisting of
very small partial areas. The surfaces exist on a flexible material and revolve on
large cylinders of which only a small part can be seen. Thus, the coincidence of
different phases of the designs on both cylinders produces a design transforming
in time. A device of this nature, the Archipenlura was presented to New York
audiences in 1931 by the sculptor Archipenko. This form offers many possibilities
through the correlation of different speeds and directions of motion (mobile
mosaics) .

"See "growth of plants." "slow motion" and 4 'Evcn when such instruments are devised, these
"accelerated motion" shown in various films. compositions might be distributed by means of

film reprints. Compare phonograph records.

rv -■ _j l r\rscs\f> Original from

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CONTINUUM

69

Art Form Number Ten (sense of sight) :

Kinetic art of visible texture transforming on a moving surface. (Art form
number seven functioning in time.)

This art form is similar to number seven in its special parameters and to
number nine in its general parameters. It is a hypothetical art form, although,
in reproduced form, it can be found in certain motion pictures. 46

Art Form Number Eleven (sense of sight) :

Static art of visible light placed inside a three-dimensional spatial form. (Art
form number seven with one additional general parameter X 3 ).

To produce the spatial form, we may use frosted glass, marble, or any other
translucent material through which the shape of the source of illumination is not
clearly visible. The resulting light may be white or colored. Many modern
lamps belong to this art form. 47 In conventional terms it might be called "sculp-
ture illuminated from within." Comparatively little of real artistic value has
as yet been produced .

Any form of illumination of an interior belongs to this group. Many ex-
amples were to be found at the Institute of Light at the Grand Central Palace
in New York. In this case the interior itself formed a three-dimensional shape,
with the spectator inside.

Art Form Number Twelve (sense of sight) :

Static art of visible pigment covering the surfaces of a three-dimensional form.
(Art form number nine with one additional general parameter X 3 ).

This art form has been known as "painted sculpture" for thousands of years
and is almost as old as sculpture itself. Many experiments in furniture making,
interior decoration, and window displays, are directed toward perfecting this art
form. In such cases, texture is often more important than color. 48

Art Form Number Thirteen (sense of sight) :

Static art of visible texture of three-dimensional forms. (Art form number
seven with one additional parameter X 3 ).

In this case, combinations of two-dimensional surfaces produce three-dimen-
sional formations:

1. Planimetric clusters.

2. Surfaces, as parts of solids.

3. Mixture of both: planimetric clusters intercomposed with solids, the

exterior of which is seen.

"The beginning of the Fall of the House of
Usher by Watson and Webber. Of course, in this
case, the illusion of moving textures is devised
optically. Ralph Steiner's Surf and Seaweed and
HtO.

47 A most effectively lighted tower is the top of
the Titania Palasl in Berlin.

**Also inlay woodwork.

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THEORY OF REGULARITY AND COORDINATION

Such an art form necessarily borrows its geometric aspect (spatial structure)
from sculpture, and its recurring patterns (texture as such), from design. It
may be described as "abstract" sculpture with emphasis on the material used.
Its highest form of development has been achieved in modern stage settings,
furniture, window displays, etc. — in other words, in applied forms.

Art Form Number Fourteen (sense of sight):

Static art of three-dimensional visible mass involving solid matter that can
retain its original form. This art can be described as a literary or "abstract"
sculpture, with emphasis on the structural expressiveness of the spatial form.
Examples are innumerable from the neolithic era to Brancusi. It consists of
planimetric systems and solids. 49

The last four art forms described become kinetic with the addition of the
fourth general parameter — time (X 4 ).

Art Form Number Fifteen (sense of sight):

Kinetic art of light projected on a three-dimensional or on a two-dimensional
screen in motion. The light source and the screen are subjected to general
parameters.

In art forms of this group, the light source changes in time continuity with
respect to its special parameters. Screens of various quality 60 and form can be
used. They may be static or kinetic." The following forms can be adopted:

1. "Quasi-two-dimensional" screens (the third dimension approaches zero:
"minimum thickness"); both sides can be used. They are planimetric
surfaces.

2. Three-dimensional spheric surfaces: both concave and convex sides
can be used .

3. Systems of screens involving planimetric surfaces, spheric surfaces, or
both (screen clusters).

4. Actual solids, showing different phases in motion, used as screens.
Human body in motion.

5. Multi-screen effects through systems of mirror reflections, and using
1,2,3 and 4 forms of screen.

Various applications of this art form can be found in the theatre and kinetic
forms of advertisement. There are innumerable instrumental possibilities in
modern engineering technique for its further evolution.

'•Bas-relief belongs here, although it is a hybrid agara Falls illuminated; "Les Fontaines Lumi-
form between sculpture and visible texture. neuscs" Paris International Exposition, 1937.

"Solid, liquid, gaseous (smoke screens). Ni- ''Moving only, or moving and transforming.

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CONTINUUM

71

Art Form Number Sixteen (sense of sight):

Kinetic art of visible pigment covering two or three-dimensional surfaces that
are in motion .

The surfaces can be classified into the same five forms as in number fifteen.
The kinetic process in a pigment can be stimulated chemically or optically."
Optical transformation has been used in many vaudeville shows: for example,
changing the color of the dancers' dresses in rhythm with the music of the dance.

Art Form Number Seventeen (sense of sight):
Kinetic art of visible texture of surface or volume.

Similar in every respect to form number sixteen in its general parameter
and based on form number thirteen in its special parameter. It would be a three-
dimensional kinetic Archipentura in a way, but with emphasis on texture. Many
industrial processes in metallurgy, in textiles, etc., suggest very powerful impres-
sions of this kind. For the geometric classification of surfaces see form number
fifteen. With the advancement of physico-chemical knowledge, the possibility
of full range variation in visible texture may be realized. 58

Art Form Number Eighteen (sense of sight) :
Kinetic art of visible mass.

This form deals with planimetric systems and volumes that move and trans-
form in time continuity with respect to one another. This art form can be de-
scribed as a kinetic abstract sculpture. It is a geometric art par excellence."
The dance belongs to this art form.

D . Time (X 4 ) as a General Parameter

The foregoing concludes the description of special parameters in the general
classification of art forms dealing with one organ of sensation at a time and with
one system of special parameters.

As to general parameters, they may be classified as follows:
Time, t — X«, is duration, an unavoidable condition underlying any real,
conditioned, or imaginary existence. It is a psycho-physical category and a
mathematical parameter. As a psycho-physical category, it has only one direc-
tion: from the past, through the present, to the future. We cannot make it run
otherwise. As a mathematical parameter it is convertible. If any event has
been recorded, it can be reproduced in all forms of changeability in time, thus
adopting the property of any other single parameter.

"There is also a mechanical possibility that can preliminary efforts in this direction can be real-
be realized through moving very small portions of ized through the cinema.

the surfaces. It would amount to something like "Solidrive, an instrument for the realization ol

three-dimensional Archipentura. Solidrama. designed by the author; a working

"The control over transformations of matter. model of Solidrive, 45* in diameter, is at the

its solid, liquid and gaseous states. At present. Schillingcr Studio.

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UNIVERSITY OF MICHIGAN

72 THEORY OF REGULARITY AND COORDINATION

The record of an event, in its special parameter of co-existing points along
the parametral extension, may be represented thus:

abcdef ghi
tj-* ■ • 1 « 1 • • a-t,

We may regard any point along this extension as the present. Then, any point
to the left will be the past and any point to the right will be the future, providing
that the direction of our movement is from a to i. 55 By inverting the direction
of movement, or by inverting the position of this parameter by means of a new
parameter, the former future may become the past, and the former past may be-
come the future. This may be illustrated in actual 5 * time by running the record
(a film or phonograph disc) backwards. By freezing portions of time continuity
in the form of recorded events, one can arrange them in new ways, which amount
to a complete transformation of the continuity of these events.* 7 This process
has been used in motion picture montage, one of the most important forms of
cinema technique. By means of this process, besides the forward-backward pro-
gression of events, all forms of continuity are obtained through permutation of
portions of time.

a b, b ►■ a

b c, c ►- b

c ► d, d c.etc,

can be used consecutively or simultaneously, thus building an entirely new tem-
poral actuality. This can be performed with any art form realized in any system
of special parameters: sound, odor, flavor, light, etc.

The relationship of psycho-physical time, and time as a general parameter
in an art continuum, may be easily illustrated through the following scheme:
If ti — t2 is a portion of time in an art continuum, and T m — T n a portion of the
general time category, then ti — 12 can be superimposed at any point on T m — T n .
Let us assume t a , t llt t r . . . as intermediate points on the parametral extension
T — T •

ta t„ t c t d t e t f

T -J • ■ ■ ■ > T

* m * n

We can superimpose ti — 12 on any point t a , t b , t c . . . For instance, we can make
t coincide with t c :

t a t b tl t2 t e tf

* ~

T

• m

"Or ti tj. '"While they arc Ikmiik reproduced in motion.

"I'sycho-physical.

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CONTINUUM

73

This means that any given art product can be performed at any time and that it
will preserve the same time relations within its own time limits (ti — ti) and regard-
less of events formerly recorded. Any work of art can be reproduced at any
time and in any place, provided that there are technical facilities.

The three parameters (Xj, X 2 , X3) of space in the world of Euclidean
geometry represent the rectangular coordinates (length, width, depth). In the
curvilinear space of immense dimensions, (Lobachevsky and Riemann), this
Euclidean space is infinitesimal. And, within this infinitesimal space, all the
Euclidean premises remain true. M In other words, there is nothing wrong with
our interpretation of space in terms of the three rectangular coordinates. We
accept this statement as a premise for the definition of spatial relations in visible
art forms.

We can produce any geometric formation with any visible art material
(light, paint, clay) by superimposing this sensuously perceptible material upon
the imaginary geometric extension. Thus, a straight line may be generated by
moving an ink point along the geometric extension of one of the parameters
(Xi, Xj, X3) •

In the previous classification of art forms, a visual art of one dimension was
not considered. In linear design, only one kind of design exists, a straight line
of any desirable length . This is true so long as time does not enter as the second
component. When time enters, the kinetic form of art gives only two very limited
possibilities:

(1) a point moving forward-backward at different velocities, and

(2) a straight line changing its dimension in time.

A line moving on a plane (behavior of Xi with respect to Xj) provides all
the forms of planimetric linear design, and also different forms of optical illusion:
perspective (suggestion of X 3 ) and the distortion of angles and dimensions.

The components of linear design on a plane, due to X lt are:

1 . dimension

2. direction

The component of linear design on a plane, due to Xi is:

3. angle

Through the combination of constant and variable dimensions, directions
and angles, an infinite variety of linear design patterns can be produced. 49

A line moving in space (behavior of Xi, with respect to Xj and Xs) provides
all the forms of stereometric linear design. 40

The components of a linear design in space due to Xi, are:

1 . dimension

2. direction

•'Einstein's general theory of relativity is re- "See Part III, Chapter 3, Design,

lated to hi* special theory of relativity as these '"Practical realization is possible through wire

two postulates of space; one is a special case of the or waxed thread. The third dimension (2r) is

other. negligible on account of its small value.

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UNIVERSITY OF MICHIGAN

74 THEORY OF REGULARITY AND COORDINATION

The component of a linear design in space due to X 2 and X 3 is the relation
of the values of angles on the two coordinate planes.

In kinetic art forms, any linear design becomes a trajectory, i.e., the trace
left by a material point moving through X lt or through XiX 2 , or through XiX 2 X 3 .

A line moving perpendicularly to its own direction forms a plane. A plane
can move with respect to Xi and X 2 , thus producing a planimetric design on a
plane. A plane can move with respect to Xi, X 2 and X 3) thus producing a plani-
metric design in space. The components of Xi, X 2 and X 3 , previously described ,
are used together with one additional component of dimension for Xt-

The kinetic aspect of a planimetric design is a planimetric trajectory (when
it moves with respect to Xi or X 2 ) or a stereometric trajectory (when it moves
with respect to X 3 ).

A plane moving perpendicularly to Xi and X 2 forms a solid. A solid can
move with respect to XiX 2 , or XiX 3 , or X 2 X 3 , thus producing a stereometric
design in space limited by two parallel planes. 61 A solid can move with respect
to XiXjX 3 , thus producing a stereometric design in space with no boundaries.™

The kinetic aspect of a stereometric design is a stereometric trajectory
(when it moves with respect to XiX 2 , XiX 3 , or X 2 X 3 ), or a hyper-stereometric 83
trajectory (when it moves with respect to X 1X2X1) .

Simultaneous combinations of spatial components here described produce
simultaneous spatial systems. The properties of such systems are:

(1) motion

(2) transformation

(3) inter-penetration 84

These are to be found in different designs as well as in relation to one another.

E . Second Group of Art Forms (More Than One Organ of Sensation
and More Than One System of Special Parameters)

In the second group of this classification, art forms previously classified
under numbers one, two, three and four do not yield any combinations since
they require more than one organ of sensation at a time.

Classifying combinations which are complex homogeneous art forms by the
number of systems representing special parameters, we obtain:

1. two systems:

light and pigment (5 and 6)

light and visible texture (5 and 7)

pigment and visible texture (6 and 7)

2. three systems:

light, pigment and visible texture (5, 6 and 7)

"Conditioned three-dimensional space. does not move with respect to spatial X4, per-

"Unconditioned three-dimensional space. pendicular to Xi. Xj and Xj.

**Some geometric schools (Hinton) consider * 4 At present they can be realized through

such a trajectory as four-dimensional solid. We multi-exposures on film.

believe it is a misconception because the XiXjXj

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CONTINUUM

75

A complex homogeneous form may consist of combinations by two or three
elements, which, in this case, are light, pigment, visible texture.

In the same way, in the kinetic group, we obtain:

8 + 9; 8 + 10" and
8+9 + 10.

In the next group:

11 + 12; 11 + 13; 11 + 14
12 + 13; 12 + 14

13 + 14 and

11 + 12 + 13; 11 + 12 + 14; 11 + 13 + 14; 12 + 13 + 14 and 11 + 12 +
13 + 14, and, in the last group:

15 + 16; 15 + 17; 15 + 18

16 + 17; 16 + 18 and
15 + 16 + 17; 15 + 16 + 18; 15 + 17 + 18; 16 + 17 + 18
and 15 + 16 + 17 + 18.

All the primary forms of each group 5 — 18 can combine with one another
by 2 to 18:

By Two:

5 + 8; 5 + 9;

6 + 8; 6 + 9;

7 + 8; 7 + 9;

5 + 10;

5 + 16;

6 + 10;

6 + 16;

7 + 10;

5 + 11

5 + 17

6 + 11

6 + 17

7 + 11

5 + 12;

5 + 18.

6 + 12;

6 + 18.

7 + 12;

5 + 13;

6 + 13;

7 + 13;

5 + 14;

6 + 14;

7 + 14;

5 + 15;

6 + 15;

7 + 15;

7 + 16; 7 + 17; 7 + 18.

8 + 11; 8+12; 8 + 13; 8+14; 8 + 15; 8 + 16; 8 + IT, 8 + 18.

9 + 11; 9 + 12; 9 + 13; 9 + 14; 9 + 15; 9 + 16; 9+17; 9 + 18.

10 + 11; 10 + 12; 10 + 13; 10 + 14; 10 + 15; 10 + 16; 10 + 17; 10 + 18.

11 + 15; 11 + 16; 11 + 17; 11 + 18.

12 + 15; 12 + 16; 12 + 17; 12 + 18.

13 + 15; 13 + 16; 13 + 17; 13 + 18.

14 + 15; 14 + 16; 14 + 17; 14 + 18.

By Three:

5 +

8+9

5 + 8 + 10

5 + 8 + 11

5

+

8 +

12;

5

+

8 + 13;

5 +

8 + 14

5 + 8 + 15

5 + 8 + 16

5

+

8 +

17;

5

+

8 + 18.

5 +

9 + 10

5 + 9 + 11,

5 + 9 + 12

5

+

9 +

13;

5

+

9 + 14;

5 +

9+15

5 + 9 + 16

5 + 9 + !7

5

+

9 +

18.

5 +

10 + 11

5 + 10 + 12

5 + 10 + 13

5

+

10 +

14;

5

+

10 + 15;

5 + 10 + 16

, 5 + 10 + 17

5

+

10 +

18.

"These numbers, suggesting various combined
arts, refer to the art forms described above (Ed.)

Digit

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UNIVERSITY OF MICHIGAN

76 THEORY OF REGULARITY AND COORDINATION

5 + 11 + 12; 5 + 11 + 13;

5 + 11 + 17;

S + 12 + 13; 5 + 12 + 14;

5 + 13 + 14;
5 + 14 + 15;
5 + 15 + 16;
5 + 16 + 17;
5 + 17 + 18.

5 + 13 + 15;
5 + 14 + 16;
5 + 15 + 17;
5 + 16 + 18.

5 + 11 + 14;
5 + 11 + 18.

5 + 12 + 15;
5 + 12 + 18.

5 + 13 + 16;
5 + 14 + 17;
5 + 15 + 18.

5 + 11 + 15;

5 + 12 + 16;

5 + 13 + 17;
5 + 14 + 18.

5 + 11 + 16;
5 + 12 + 17;
5 + 13 + 18.

Combining 6 with the rest by Three 66

( 6 + 8) + 9, + 10, + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18.

( 6 + 9) + 10, + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18.

( 6 + 10) + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18.

( 6 + 11) + 12, + 13, + 14, + 15, + 16, + 17, + 18.

( 6 + 12) + 13, + 14, + 15, + 16, + 17, + 18.

( 6 + 13) + 14, + 15, + 16, + 17, + 18.

( 6 + 14) + 15, + 16, + 17, + 18.

( 6 + 15) + 16, + 17, + 18.

( 6 + 16) + 17, + 18.

6 + 17 + 18.

Combining 7 with the rest by Three :

( 7 + 8) + 9, + 10, + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18.

( 7 + 9) + 10, + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18.

( 7 + 10) + 11, + 12, + 13, + 14, + 15, + 16, + 17, + 18.

( 7 + 11) + 12, + 13, + 14, + 15, + 16, + 17, + 18.

( 7 + 12) + 13, + 14, + 15, + 16, + 17, + 18.

( 7 + 13) + 14, + 15, +' 16, + 17, + 18.

( 7 + 14) + 15, + 16, + 17, + 18.

( 7 + 15) + 16, + 17, + 18.

( 7 + 16) + 17, + 18.

7 + 17 + 18.

Combining 8 with the rest by Three:

( 8 + 11) + 12, + 13, + 14, + 15, + 16, + 17, + 18.

( 8 + 12) + 13, + 14, + 15, + 16, + 17, + 18.

( 8 + 13) + 14, + 15, + 16, + 17, + 18.

( 8 + 14) + 15, + 16, + 17, + 18.

( 8 + 15) + 16, + 17, + 18.

( 8 + 16) + 17, + 18.

8 + 17 + 18.

"These numbers, suggesting various combined
arts, refer to the art forms described above (Ed.)

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CONTINUUM

77

Combining 9 with the rest by Three:

( 9 + 11) + 12, + 13, + 14. + 15, + 16, + 17, + 18.

( 9 + 12) + 13, + 14, + 15, + 16, + 17, + 18.

( 9 + 13) + 14, + 15, + 16, + 17, + 18.

( 9 + 14) + 15, + 16, + 17, + 18.

( 9 + 15) + 16, + 17. + 18.

( 9 + 16) + 17, + 18.
9 + 17 + 18.

Combining 10 with the rest by Three:"

(10 + 11) + 12, + 13, + 14, + 15, + 16, + 17, + 18.
(10 + 12) + 13, + 14, + 15, + 16, + 17, + 18.
(10 + 13) + 14, + 15, + 16, + 17, + 18.
(10 + 14) + 15, + 16, + 17, + 18.
(10 + 15) + 16, + 17, + 18.
(10 + 16) + 17, + 18.
10 + 17 + 18.

Combining 1 1 with the rest by Three :

11 + 15 +16; 11 + 15 + 17; 11 + 15 + 18.
11 + 16 + 17; 11 + 16 + 18.
11 + 17 + 18.

Combining 12 with the rest by Three :

12 + 15 + 16; 12 + 15 + 17; 12 + 15 + 18.
12 + 16 + 17; 12 + 16 + 18.
12 + 17 + 18.

Combining 13 with the rest by Three:

13 + 15 + 16; 13 + 15 + 17; 13 + 15 + 18.
13 + 16 + 17; 13 + 16 + 18.
13 + 17 + 18.

Combining 14 with the rest by Three :

14 + 15 + 16; 14 + 15 + 17; 14 + 15 + 18.
14 + 16 + 17; 14 + 16 + 18.
14 + 17 + 18.

Combinations by four and more elements can be obtained by the process
described below.

M These numbers, suggesting various combined
arts, refer to the art forms described above (Ed.)

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78

THEORY OF REGULARITY AND COORDINATION

F. Third Group of Art Forms (More Than One Organ of Sensation and
More Than One System of Parameters.)

Art forms numbers one, two, three and four combined with one another or
with any of the remaining (5 — 18) forms belong to this group.

Classifying combinations, which are complex heterogeneous art forms by the"
number of organs of sensations required, we obtain:

1 . Two Organs of Sensation :

Sound and texture
Sound and odor
Sound and flavor
Texture and odor
Texture and flavor
Odor and flavoi

2. Three Organs of Sensation :

Sound, texture and odor
Sound, texture and flavor
Sound, odor and flavor
Texture, odor and flavor

3. Four Organs of Sensation:

Sound, texture, odor and flavor (1 + 2 + 3 + 4)

A complete heterogeneous form may consist of combinations by two, three
and four elements, which in this case are sound, texture, odor and flavor.

The rest of the complex heterogeneous art forms is obtainable from the
simple form 5 — 18 combined with forms 1, 2, 3 and 4 by two. To combine 3, 4,
5 — 18, use complex homogeneous forms and combine them with forms 1,2,3 and 4.

For general orientation in finding any desirable combination by any number
of elements, the chart on page 79 is supplied. To find any combination of the
forms represented by consecutive series, or partly consecutive series, use one
direction. If the numbers cease to be consecutive, move perpendicularly to the
previous direction, until the desired number occurs. Always move down or from
left to right. The proper method for discovering combinations is by underlining
the path at the bottom of a number for the horizontal progression, and at the
left side of the number for vertical progression .

(1 and 2)
(1 and 3)
(1 and 4)
(2 and 3)
(2 and 4)
(3 and 4)

(1+2+3)
(1+2 + 4)
(1+3+4)
(2 + 3+4)

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

CONTINUUM

79

X,X t X s

X^XjX,

/

*

3

* *

—Aa

Sr

6

/

S
' 8

9

'*/

'll

IZ

13

It

/

■ A

' IS

1*

*

3

*>

'$-

*

7 /

't

9

K> ,
/

-7* —

'//

/z

13

If

17

//

3

* -

/

' r

6

-7—

' a

9

'V

/ ll

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73

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-V—

'is

/«

'7

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7 ,

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6

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17

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r/-

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17

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-A-

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78 .
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/

The diagonal lines indicate borders between the groups classified through
the number of general parameters.

X 4 = time X 3 = depth X 2 = width Xi = length

Figure 10. Chart for the combination of complex art forms.

Digiti

Google

Original from
UNIVERSITY OF MICHIGAN

80

THEORY OF REGULARITY AND COORDINATION

G. Correspondences Between Art Forms

We shall conclude our classification of art forms with an examination of the
correspondences among different art forms.

Two kinds of correspondences are possible:

(1) absolute correspondence of values in terms of measurement.

(2) absolute correspondence of relations in terms of measurement.

Both of these kinds may be either subjective or objective, personal or impersonal.

A great many tests have been performed and a number of books written on
the subjective correspondences between color and sound, odor and sound, and even
flavor and sound. Some individuals, with so-called "color audition," insist that
the pitch of a certain tone is green, another, red, etc. Similar associations have
been obtained in relation to musical chords and even tonalities. Thus, some
people believe that F sharp major is purple. With reference to tone quality,
they consider the oboe tone quality a dark orange and the trumpet, a bright
yellow. In Huysmans' novel, A Rebours, the hero, DesEsseintes, classifies his
liquors by associating them with the tone qualities of different musical instru-
ments, and labels them "flute," "clarinet," etc. There are also subjective asso-
ciations with temperature. Some people consider c minor cold, and d minor
warm, while E major is supposed to suggest the temperature of a flame.

All these associations are different with different individuals and therefore
cannot be obligatory for anyone else. Subjective correspondences of values
reveal no definite system of relations with physical values (intensity, frequency)
forming an excitor.

Objective correspondences of physical frequencies or intensities, adopted by
many authors of color audition theories, establish correlations between the low-
est audible pitch (frequency of 16 periods per sec.) and the shortest visible wave-
length (.00072 mm.) 64 . Then the authors assume that the highest audible pitch
must correspond to the longest visible wave-length. In some cases, the corres-
pondence is built on an attempt to correlate the 2:1 ratio of sound wave frequen-
cies (an "octave" in music) with the whole visible spectrum, thus associating all
the hues with one octave in pitch. In order to derive other color values corres-
ponding to other octaves of pitch, intensification (increase of luminosity) of the
same progression of hues is applied.

These systems of absolute correspondences of values in frequencies and
intensities are arbitrary and pseudo-scientific. Firstly, the approximate ratio
of pitch relations is 1:1000 while that of the visible spectrum equals approxi-
mately 7:4. Secondly, the elementary ratio of pitch relations is 2:1 while the

"Robert W. Wood. Physical Optics.

1 . Subjective Correspondences of Value

Objective Correspondences of Values

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UNIVERSITY OF MICHIGAN

CONTINUUM

81

ratios of primary colors are very different. The approximate ratio of blue and
green equals 9:10; green and red 70 equals 10:13; blue and yellow equals 3:4; yel-
low and red equals 7:8. Thirdly, no natural association has been observed which
would apply to everyone. Thus, the correspondence of an extremely low red
with the tone "c" of 16 vibrations per second cannot be enforced as a law.

3. General Correspondences of Relations

According to the geometric concept of extension, one can insert as many
points as are desired between two given points, "a" and "b." In terms of recti-
linear measurement, this amounts to dividing length into a number of uniform
units (linear values). In terms of circular measurement, these uniform units
may be represented through angular values (degrees, it). In order to secure a
uniform scale of units between two given limits, it is necessary to determine the
ratio of the limits in terms of frequencies. If a rect ilinea r or a circular extension
is the graph of a special parameter, one can determine the scale of units in actual
frequencies, such as pitch or hue .

Algebraically it may be expressed as follows:

where "u" is a unit in terms of frequencies (oscillations) ; -fr-, a ratio expressing
the limits in terms of frequencies; exponent "n,"— the number of un i ts required
in a scal e. This formula has general meaning and may be applied t o any param-
eter of art materia l. 71

One of t he well-known applications of this form ula is in the syste m o f the
equal ly tempered scale of tonal pitch where the u nit called the semitone is:

The numeral 2 indicates the 2:1 ratio of vibrations (256:128, for instance), and
t he number 12, the number .of such_ units between one and two.

On a logarithmic rectilinear graph of pitch values, this system of units will
consist of twelve equidistant points between logarithm 1 and logarithm 2, if the
base of the logarithm equals y/2. On a circular graph serving the same pur-
pose, there would be 12 equidistant points on the arc between its extreme points.

base V^", thus g = b.
V'HT' V~W(= a) (Ed.)

Original from
UNIVERSITY OF MICHIGAN

7B This is the approximate ratio of complemen-
tary colors.

"The scale of units y/ ^ can be graphed
as the scale of logarithms from ba to a. to the

Digitized byGoOQ.G

82 THEORY OF REGULARITY AND COORDINATION

The intermediate points can be found by measuring the segments of the arc
through corresponding angles.

Figure 1 1 . The twelve divisions represent the equal temperament tuning system .

120°

A = 120° = ^jt = 10°

ob = 120°

o

Figure 12. Tuning system as represented in 120° of a circular graph.

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UNIVERSITY OF MICHIGAN

CONTINUUM

83

Thus, the relations of one parameter may be made to correspond with the
relations of another. In a musical composition, transposition made from one
tonal base to another changes the general quality only, without affecting the
pitch-time-intensity relations. It is considered the same composition appearing
in a different form of instrumental sonority. The essential components are
pitch, time and intensity in their relations, perceived as different degrees of
similarity and contrast. In the same sense, the relations of hue, intensity and
saturation in color admit transposition from one color base to another. And
in the same sense, these relations are perceived as degrees of similarity and
contrast.

jWith such a premise, a given portion of one parameter, split into"n" units,
will adequately correspond to any chosen portion of another parameter split
into the same number of units. Thus, for example, once we establish a corre-
spondence between the equally-tempered twelve-step scale of pitch units and the
full range of visible hues arranged as a uniform progression of twelve color-units,
we can transcribe a musical melody into a color progression. The absolute value
of a tonal base and the absolute value of a color base, with respect to the tonal
base, is not essential. The correspondence of relations in two systems is essen-
tial. The tone "c" as a tonal base might be used with any color base, green,
yellow, red or other. But as long as "c" moves to "e flat," for example, the
selected color base must change adequately, i.e., on 90° of the spectral circle."
The color base being yellow, for example, will change to orange (0)> or to
green (O) "") •

We shafldetermine correspondences as being normal through the probability
of their recurrence and through their pragmatic physical adequacy." Such cor-
respondences are predominant in art. The improbable correspondences serve
the purpose of distortion and exaggeration.

All correspondences realized through associations, "normal" as well as
"abnormal," are due to conditioned physiological reflexes caused by the physical
excitors affecting an organ of sensation. All the problems of artistic taste must
be studied from this angle. "Beauty" is a psychological complex and a derivative
of physiological reflexes. It is a form of satisfaction induced by a relation or a
system of relations of simultaneous or consecutive excitations. In order to pro-
duce an effect of "beauty," the percentage of excitations previously experienced
must be quite high. Insofar as the normal musical taste of the trained majority
is concerned, for instance, the amount of familiar excitations should be quite
highy about 85% or more.

\One of the necessary moments in artistic enjoyment is a standard deviation
f rom the nearest simple relation. For example, if a progression of two uniform
durations becomes too obvious, human intuition finds a value of standard devia-

n "c" to "eV" represents one-quarter of an oc-
tave, or one-quarter of 360° = 90°. This can be
performed either clockwise or counterclockwise.
It is also possible to establish and use a system of
secondary relations such as: PiPy = n:mn.
All the correspondences may also be used in oblique
and inverse relations. (Ed.)

Digitized byGoOgle

7, An exampJe of normal correspondence: A
giant (dimension) must be heavy (gravity), speak
with a bass voice, (dimensions of the vocal cords)
and possess red cheeks (coloration as a result of
good blood circulation) . By changing some of
theBe components, one can easily illustrate an
abnormal correspondence.

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84

THEORY OF REGULARITY AND COORDINATION

tion. For example, instead of performing 6 J exactly (as£ +$) American "croon-
ers" will sing d/W. (which is -f- +"§■), ord.^J (which is-$- +-&). One eighth, in
this case, will be a standard deviation, an "artistic infinitesimal ."7

From a mathematical viewpoint, beauty may be expressed as a differential
variation of a rational term (a relation, or a system of relationships), where the
rational term and the differential derive from one homogeneous harmonic series.
T he rational term is usually a commonly known idiom, and the differential
( artisticdifferential") comes as a further refinement ^ "

Esthetic satisfaction comes mainly from the sensation of being off balance,
but in an obvious relation to balance. Here a mechanical experience becomes
an artistic one through the discrimination of the "artistic differential" by our
senses. Then the joy of discriminating simple relations in a sensory form possess-
ing a standard deviation is psychologically similar to solving problems or riddles.
The element of the unknown stimulates curiosity, and the process of associating
it with the known produces a feeling of satisfaction. Here lies the success of one
■work of art and the failure of another. This explains why a jig-saw puzzle hobby
may become an epidemic. 74

T4 See "Periodicity of Expansion and Contrac-
tion" in Chapter 3. Periodicity.

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CHAPTER 2

CONTINUITY

f~*l ERTAIN points in a continuum sometimes acquire special significance.

This happens either because they form a class, or because they happen to
belong to a particular class. By a particular class, we mean one that is simple
or axiomatic, such as the class of natural integers; or, one that involves some
harmonic relationship, such as a class that develops according to a constant
ratio: a geometric progression, a summation series, etc.

Continuity is a finite portion of continuum. It is formed by the progression
of extensions along any coordinate or parameter. The terms of such progression
are expressed through the relations of their numerical values. The manner of
transition (gradual or sudden, through rational numbers or differentials) from
the antecedent to the following term in a continuity is dependent upon the rela-
tion of the values determining these terms, and upon the value of the differential
between the terms. The relative values are determined by the terms of periodicity
of different forms and orders.

A. Series of Values

A continuity consists of coordinate and parametral components. Parametral
components determine coordinate components in sensory forms. Parametral
extensions are the positive values. Coordinate extensions as such are the nega-
tive values.

Both positive and negative values produce a series based on one or another
form of regularity. The type of series upon which a certain continuity is based
determines the potential forms of development and growth of such continuity.

1. Natural Series.

Integer Series — A natural integer series is an infinite progression of all
the integer numbers between one and any integer value approaching infinity.

n

2, = 1,2,3, n. There can be finite portions of a natural integer series

i

between two given limits.

Thus, a series from 1 to 5 will be:

6

£ = 1,2, 3, 4, 5 or generally,

i

ID

m

X = a, b, c,

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UNIVERSITY OF MICHIGAN

86 THEORY OF REGULARITY AND COORDINATION

If we plot a curve through points that are the terms of a natural integer series,
its curvature will approach zero and will reach zero at the infinity point of the
series. Therefore, the degree of curvature is in inverse proportion to the number
value of a plotted point reached by the curve. The degree of curvature will be
greater at the point p = 2 than at the point p — 7.

Fractional Series — A natural fractional series is an infinite progression of
all the rational fractions with the numerator one and denominators with any
integer value.

L ~ V 3' 4' n

2

where n approaches infinity.

Thus a series from 4- to \ will be:

4-

L, = "2' J- J' y or g eneral, y.
T

J_

m

V -ill 1

j a b c m

"a

If we plot a curve through the points which are the terms of a natural fractional
series, the degree of curvature of such a curve will approach zero, and will reach
zero at the point -J5 of the series. The degree of curvature will increase in its
first few terms and decrease to zero with each succeeding term.

rv m C^nr\cs\£> Original from

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Figure 3. Natural integer and fraction series plotted together.

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88 THEORY OF REGULARITY AND COORDINATION

2. Other Series.

Various other series of integral or fractional values can be formed, though
these series are not "natural."

(1) Arithmetical Progression Series. A series whose values increase in
an arithmetical progression, such as:

x+mc

£ +c = x, x + c, (x + c) + c, (x + 2c) + c, + (x + mc)

x

If x = 3 and c = 2, then:

3+2m

X +2 = 3, 3 + 2. (3 + 2) + 2, 3 + (2 X 2) + 2,

3

. . + 3 + 2m = 3, 5, 7, 9, 3 + 2m

The same values may be used as denominators for the fractional series.

i+mc

y 1 1 i i i

~+ c x' x+c' (x+c)+c' (x + 2c)+c' " ' " (x+mc)

x

1

3+im

Z, +2 3- 5- y 9 3+ 2m

3

(2) Geometrical Progression Series. A series whose values increase in a
geometrical progression, such as:

c m x

X i CX fOXj • * • * C X

X

If x = 3 and c = 2, then

2 m x

2 2 = 3, 3X2. 3X2 1 , . . . 3X2" = 3, 6, 12, . . . 3X2"
The same values may be used as denominators for the fractional series.

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CONTINUITY

89

1

-m,

1

T

3x

c=x

z

J_ 1 1 1

x' cx' c*x' ' ' ' c m x

v m _L_L _L J_

1,2 3' 6' 12' 3x Im
+

When x = c, then:

x, x 2 , X s , ... x m

This last is the most frequent form of musical rhythm.

(3) Summation Series ("Fibonacci Series"). A series in which each
succeeding term equals the sum of the two preceding terms.

Xs = x, y, x-f y, x+2y, 2x+3y 3x+5y, 5x+8y

Example:

55

2 s = 1, 2, 3, 5, 8, 13, 21, -34, 55

i

This series is the foundation of the theory of Dynamic Symmetry evolved by
Jay Hambidge and his group. Hambidge found that this series had been applied
in ancient Egyptian and ancient Greek art. This series was also observed by
Professor Church of Oxford as the mechanical basis of the growth of certain
plants.

Here are various summation series:

1, 2, 3, 5, 8, 13, 21

1, 3, 4, 7, 11, 18

1, 4, 5, 9, 14, 23

1, 5, 6, 11, 17, 28

1, 6, 7, 13, 20, 33, . . .

1, 7, 8, 15, 23, 38, . . .

1, 8, 9, 17, 26, 43, ...

1, 9, 10, 19, 29, 48, . . .

1, 10, 11, 21, 32, 53, . . .

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90

THEORY OF REGULARITY AND COORDINATION

2, 4, 6, 10, 16, 26, . . .
2, 5, 7, 12, 19, 31, ...
2, 6, 8, 14, 22, 36, ...

2, 7, 9, 16,25,41, ...

* * * *

3, 6, 9, 15, 24, 39, . . .
3, 7, 10, 17, 27, 44, . . .
3, 8, 11, 19, 30, 49, . . .
3, 9, 12, 21, 33, ...

The inverted fractional form is:

V J. J_ _1 1 1

x'v'x-

y'x+y" x+2y' 2x-f-3y" " "

V = !J_J_± J_

s 2' 3' 5' 8' 13

(4) Series of Natural Differences. A series in which the difference
between preceding and succeeding terms grows in the natural integer series-

£ D = x,x + l, (x+D+2, (x+3)+4, ...

37

£ D = 1, 2,4, 7, 11, 16, 22, 29, 37

i

(5) Series of Prime Numbers. A series in which the whole progression
consists of consecutively increasing prime numbers.

If x, x 2 , x 3 .... are the prime numbers arranged in increasing values, we
may write:

x l i *2, X 3' • • • ■ X n

"1

23

£ pn = 1, 2, 3, 5, 7, 11, 13, 17, 19, 23

i

rv -■ _j l nnnlp Original from

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CONTINUITY

3. Rational Ratios of the Natural Series Fractional Continuity.

2n is any even number place.
2n + 1 is any odd number place.

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92

THEORY OF REGULARITY AND COORDINATION

1 1
m — 1 "* — m —

. ^ is the relation of distances of two con-

m ~* — *~ m— 1

secutive terms of fractional continuity.

k is any numerator of the ratio taking an odd place in the series,
ki is any numerator of the ratio taking an even place in the series.

Thus, the consecutive division of a rectilinear segment produces a complex alter-
nating series of fractional values.

Terms in the odd places form:

i

V .2-1-1 m

2,+, ji 2 ' 3 " • • m _!

Terms in the even places form:

V 111 _n_

^+2 3' 5' 7"" n-2

Then the whole series takes the following form:

v ill 7 ! 9 jn_ _n_

2/+I.+2 r3'2'S'3'7""m-l'n-2
+

I 7 i f V J *
H 1 1 1 1 1 1 1~

Figure 4. Natural series fractional continuity.

B. Factorial- Fractional Continuity

May be represented through normal series with any desirable number of
places by filling out the intervals between the already existing terms of a series.

The zero series has two terms:

cc

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UNIVERSITY OF MICHIGAN

*

CONTINUITY 93
The first series has three terms:

x

... 00

X

I, -o

The second series has five terms:

= ° i-.-i.-.x... CO
The third series has nine terms:

^ - g . i-

in>©>l « «

X X

. . . x" . . . 00

Any value, integral or fractional, may be attributed to x.
Example:

A five term series, when x = 5

X 2 x=s - 0. . 5 ... oo

5 5

r

If the limits are given, for instance, between — and x n , then:

x

(i) n

If n = 3, then:

(I) 3

The middle term of the first series, if the latter is a normal one, is unity;
the value attributed to its numerator and denominator will determine the suc-
ceeding development of the series, provided that the series is a homogeneous one.
Thus, I is the determinant of a series.

The values on the left side of a series determine the terms of fractional con-
tinuity, which will be designated by "f". The values on the right side of a series
determine the terms of factorial continuity, which will be designated by "F".

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94 THEORY OF REGULARITY AND COORDINATION

In a normal series the product of a term of F by a term of f , if they are equi-
distant from the middle term, is unity.

F„f„ = 1

Therefore, the relation of equidistant terms of Factorial and Fractional continuity
can be expressed through

F n =7- and

If F n = x 3 , then: f n = -^j , and vice versa.

F = 1- — - = x l

n x s X

When x = 3

f =1 = 1= JL
n F n x s 9'

then the three term series will be:

x3

1*=3
3

= -k- . . ■ 4- • • 9

C. Elements of Factorial- Fractional Continuity Expressed through
Series

1. Monomials.

00

2.. -

...i

x

00

00

X 2* =

o...i...i

X X

00

2- f..f-- - - --

rv -r **i K fnnrilf> Original from

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CONTINUITY
2. Binomials.

95

o x +y

oo

£ 2 «+y = . . . -rJ- . . . ... x+y ... *

„ x+y x+y

_ ...(+.y ...(4-Y.. ' ...5+i

o \x+y/ \x+y/ x+y x+y

... x+y ... (x+y) 2 ... (x+y) n . . . oo

J. Polynomials.

00

V « a+b+c+...+m

A i»+b+c+. +m = 0. . . n— ; ; ; ...»

"i+b+c+. . . + ~

m

v a 1 a+b+c+ . +m

2, 2 . +b+c+ ... +In = o... a+b+c+ +m ... a+b+c+ +m .

. . . (a+b+c+. . . +m) ... °o
2 3 .+b+c+...+m ~ • • \ a +b+c+. . .+m/ *•'

/ 1 V 1 a+b+c+ . . . +m

\a+b+c+. . . +m/ ' ' "a+b+c+. . . +m" ' ' a+b+c+ . . . +m"

. .(a+b+c+. . .+m) ... (a+b+c+. . . +m)».
. .(a+b+c+. .+m) n ... oo

x n (x+y) n (a+b+c + . . +m) n

Formulae for £ , £ , and Y

(±) n (rk) n (.+bU.-u ) n

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96 THEORY OF REGULARITY AND COORDINATION

mn ( x ) " " x" ' x " ' " X

W

x* ... x n

(a+b+c+ +m) n / 1 V / 1 V

2 = Va + b+c+ . +m/ " \a+b+c+. . . + m/

(a+b+ci. . +ra) n

1 a+b+c+ ■ ■ ■ +m

a+b+c+ . . . +m ' '. ' a + b-f-c-f- . . . + m ' '

. . . (a+b+c+ . . . +m) . . . (a+b+c + . . . +m) 2 . . . (a + b+c + . . . + m) n

4. Binomial Series. — x = a + b.

y = + -JLY ( a 1 b V _2_ + _L

ZT +b Va+b^a + b^ 'Va + b^a + by/ " 'a+b + a+h '

a+b
a+b' - '

a+b . . . (a+b)* . . . (a+b) n

5. Polynomial Series. — x = a+b+c+ . . . +m.

A *=a+b+c+. +m = ( j-j-- 7 1 iT" i , h

^ n k a + b + c+ . . . +m a -f l> + c + . . fm

+ ... . m . y.

a + b+c+...+m a+b+c+ . . . + m,

( a , b , c 4.

\a + b+c+ . . . +m T a+b + c+. . . +m a + b+c+ . . +m T

+ a + b+c+ . . . +m ) ' ' ' U+b+c+ . . . +m + a + b+c+ . . . +m +

c ^ _^ m a + b+c+...+m

a + b+c+...+m "" a + b+c+. . .+m/ ■ a + b+c+ . . . +m" '
. .(a+b+c+. . . +m) ... (a+b+c+. . . +m)» ... (a+b+c+ . . . +m) n

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CONTINUITY
6. Series Consisting of Positive and Negative Values.

97

y -x"... -x...-:... -i Y-iY. .(-L)°...

-£ i* X X \ X/ \ x/

-SO

...o...ay...(iy...i..i...x...

\x/ \x/ X X

jr ... x

In a factorial-fractional continuity of rational values, the exponents are in
inverse proportion to their bases.

Averages observed in music of the civilized world:

Bases

1_

2

Exponents
4

Values for Factorial and
Fractional Continuity

# # # 4 i t **f * # i

I:

7

1

T

5
1

T

2
I

Figure 5. Averages in music of the civilized world.

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UNIVERSITY OF MICHIGAN

98 THEORY OF REGULARITY AND COORDINATION

7. Series of Averages Observed in Music of the Civilized World. 1

8. Some of the Series Observed in Folklore.
(j) ' ' (3") . . . -j. . .-|-. . .3 . . 3 s . . . . (American Indians) 1

. . -jXy. ■ *T- • • ■ - 5- • • -2X5. . (Great Russians)

D. Determinants

The value of any term in a factorial-fractional series with one determinant
can be expressed through:

F„- = x n

F n « represents a factorial unit of the nth place in the series, counting from the

initial unit — , the determinant of the series; f n « represents a fractional unit of

x

the nth place in the series, counting to the left from the initial unit — .

, „. , tJ . , t . \- .l of both these quantities, the measure-rhythm be-

'The fractions to the left ofr- represent rhythms , . .7 . . ' , .. , ' . .,

t K ing structurally related to the phrase-rhythm.

within measures while the numbers to the right (Ed.)

represent the rhythms of the measures themselves. *Helen Roberts. Forms of Primitive Music
To Schillinger, musical form is an integrated unity (American Library of Musicology, 1932), page 39.

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CONTINUITY

99

The consecutive terms of a series, following to the right from the initial term

x

— produce units of factorial periodicity in their increasing orders. The consecu-
x i

x

tive terms of a series, following to the left from the initial term—, produce units
of fractional periodicity in their increasing orders. Thus, the value n expresses:

(1) the order of a unit in a series;

(2) its place in a series;

(3) the power index of a base x.

In order to find the value of x at a given order n, it is sufficient to raise x
to the nth power.

F n ° = x n « = x", because:

x. . . x 2 . . . x" . . . x" [powers]

1° 2° 3° n° [orders]

/l\" /A"
Likewise. f n » = I — J = 1—1 , for:

(7) " • (7) (I) 7 [powersl
n° 3° 2° 1° [orders]

Example:

x 2 1

— = — . Find the value of x and of at their fourth order,
x 2 x

F 40 = 2* = 16

The determinant of a series in its factorial-fractional form may be expressed
through the following identity:

F l4 f,. = T = — = x (^-) =— + — + ... + —

X \X/ X X X

and the general form of a factorial-fractional unit of the nth order:

F - " T " " (f)" =7+£+

If the determinant of a series is m, and x ^ m, then the orders grow through
the powers of m.

Fl . f, - T = n,(f) = [m*(i)] = m (^ + «J+ +h) and

F„.f„.= T„> m "(i)= m »(| + |+.
Here, m is the coefficient of growth of the term x.

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100 THEORY OF REGULARITY AND COORDINATION

Example:
x = 3; n ■■

= 4

T - 1-
T " 3 "

3 V3y- t + j + t

T 4 i. = T 4

x = 3; m

= 2; n = 4

K<t)]-<* + M)

T<* - 2 4 (

81

When it is desirable to plan a continuity in advance, with respect to the
number of units and their values at a given order n, the following formulae should
be used:

1. Series With One Determinant.
(a) m = x

T n . = T n = x(x"- 1 ), x J (x n - 1 ), x^x"-*), . . . x n - 3 (x a ), x n - J (x J ), x n ~ 1 (x).

The continuity T at the nth order equals T to the nth power, equals x units
of the value x n_1 , x 2 units of the value x n_s , etc.

Examples:

x = 2; n = 5; T 6 = x 5 = 2 6 = 32
T 5 o - T* - 2(2*), 2 2 (2 3 ), 2»(2«), 2«(2) = 2(16), 4(8), 8(4), 16(2).

x = 3; n = 5; T» = x 6 = 3 s = 243
T 5 . = T 6 = 3(3*), 3 2 (3 3 ), 3 3 (3 2 ), 3 4 (3) = 3(81), 9(27), 27(9), 81(3).

x = 4; n = 5; T s = x* = 4 s = 1024
T 5 . = T 8 = 4(4*), 4 2 (4 3 ), 4 3 (4 2 ), 4 4 (4) = 4(256), 16(64), 64(16), 256(4).

x = 5; n = 4; T 4 = x 4 = 5 4 = 625
T 4 . = T 4 = 5(5 3 ), 5 2 (5 2 ), 5 3 (5) = 5(125), 25(25), 125(5).

x

7 ; n = 4; T 4 = x 4 = 7 4 = 2401

T 4 o = T 4 = 7(7 3 ), 7 2 (7 2 ), 7 3 (7) = 7(343), 49(49), 343(7).

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CONTINUITY

101

'M

i

i

>•

r <r'

Y' T

<

rff

'6

t

!

■

<

L

<

:

:

!

■

t

*

-

p

H

:

E

/'

*

■

Y

A

■

-

•

If

z*

+

i

I

3*

Z

J

/

J

z

r*

Figure 6. Series with one determinant.

(b) m ^ x

T n « = T" = mx(m il-5 x), m^(m n_J x), m s xCm°- 3 x), . ..
. . . m n - 3 x(m 3 x), m^Vfin^), m n - 1 x(mx).

The continuity T at the nth order equals T to the nth power, equals mx
units of the value m"^x, m J x units of the value m""*x, etc.

Examples:

x = 3; m = 2; n = 5; T* = m^x 1 = 2* X 3 s = 288
T a . - T» - 2X3(2«X3). 2 S X3(2*X3). 2»X3(2*X3), 2<X3(2X3) = 6(48),
12(24), 24(12), 48(6).

x - 2; m = 3; n = 4; T* = m'x 1 = 3 4 X2* = 324
T 4 . - T* - 3X2(3 S X2), 3*X2(3»X2), 3 s X2(3x2) - 6(54), 18(18), 54(6).

x = 5; m = 2; n = 4; T* - mV ~ 2<X5 J = 400
T 4 * = T< = 2X5(2 3 X5), 2 I X5(2 a X5), 2*X5(2X5) - 10(40), 20(20), 40(10).

■

2. Series With Two Determinants.
x and y are the two periodically alternating determinants of the series,
(a) The order of T is 2n + 1 .

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102

THEORY OF REGULARITY AND COORDINATION

T 2n +i = x(x n y n ), xy^y- 1 ), x*y (x'-'y"- 1 ), x"y* (x'-'y-'), ...
% . . x n_1 y n-I (x s y s ) i x n_1 y 0-1 (x*y) , x n y n_1 (xy) , x n y n (x) .

Example:

x = 2; y = 3; 2n + 1 = 7

T 7 = x(x s y s ). xy(x 3 y s ). x*y(xV). xV(xV). xV(xy), xV(x) = 2(2 3 X3 3 ),
2X3(2 3 X3 J ), 2*X3(2 J X3 l ), 2 I X3 I (2 l X3), 2 3 X3 l (2X3), 2 3 X3 3 (2) = 2(216),
6(72), 12(36), 36(12), 72(6), 216(2).

(b) The order of T is 2n.

T 2n = x(x n y n - 1 ), xy(x n - , y n - 1 ), x , y(x n ~ 1 y n ~ , ) 1 x n - I y n_J (x I y).

x n-i y n -»(xy), x n y n_1 (x).

Example:

x = 2; y = 3; 2n = 8

T 8 = x(x 4 y s ). xy(x s y J )» x l y(x , y I ). x*y*(x*y*)» x 3 y l (x*y). x J y s (xy), xV(x) =
2(2<X3 3 ), 2X3(2 3 X3 3 ), 2 J X3(2 3 X3 l ), 2 J X3*(2*X3 S ), 2 3 X3 2 (2»X3), 2 3 X3 3 (2X3),
2 4 X 3 3 (2) = 2(432), 6(216), 12(72), 36(36), 72(12), 216(6), 432(2).

When m is the determinant of a series and x and y are periodically alternating
coefficients of growth, m enters as a factor in both co-factors.

T 2n +i = mx(mx n y n ), mxy(mx n y n - 1 ),

T 2n = mx(mx n y n_1 ), mxy(mx n - , y n_1 ), ....

Example:

x = 2; y = 3; m = 5; 2n + l = 7
T 7 = 10(1080), 30(360), 60(180), 360(30), 1080(10).

x = 2; y = 3; m = 4; 2n = 6
T 6 = 8(288), 24(144), 48(48), 144(24), 288(8).

3. Series With Three Determinants.
x, y and z are the three periodically alternating determinants of the series,
(a) The order of T is 3n-f 1

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103

T 3n +i = x(x"y n z n ), xy(x n y n z n - 1 ), xyz(x n y n -'z n - 1 ), x*yz(x n " l y n " ,zn " 1 ).

xVz0c n_1 y n_1 z n " 1 ). xVz 2 (x n_1 y n " Izn " 2 ). x n - 1 y n ~ 2 z n_1 (x*y 2zI ).

x n-i y n-i z n-i (x^z) , x n_1 y "-^"-'(x'yz) , x n y n-1 z n_1 (xyz) , x n y n z n - 1 (xy),

x n y n z n (x).

x = 2; y = 3; z = 5;3n + l = 7

T 7 = 2(2 2 X3 2 X5 2 ), 2X3(2 2 X3 2 X5), 2X3X5(2 2 X3X5), 2 2 X3X5(2X3X5),
2 1 X3 I X5(2X3), 2 2 X3 2 X5 2 (2) = 2(900), 6(180), 30(60), 60(30), 180(6), 900(2).

(b) The order of T is 3n.
T 3n = x(x n y n z n - 1 ), xy(x n y n -'z n - 1 ), xyz(x n - 1 y n " 1 z n_1 ). x , yz(x n - l y n - 1 z n - , ) )

x , y 2 z(x n_1 y n " 2 z n " 2 ) . x 2 y 2 z 2 (x n - 2 y n_2 z n - 2 ) x n - 2 y n - J z n - 2 (x J y 2 z 2 ) ,

x"- 1 y n - 2 z n - 2 (x 2 y 2 z), x n - 1 y n_1 z n - 2 (x 2 yz), x n -'y n - 1 z n - I (xyz), x n y n - 1 z n - , (xy),
x n y"z n_1 (x).

Example:

x = 2;y = 3;z = 5;3n = 6

T 6 = 2(2 2 X3 2 X5), 2 X3(2 2 X3 X5), 2 X3X5(2 X3 X5),

2 2 X3 X5(2 X3), 2 2 X3 2 X5(2) =2(180), 6(60), 30(30), 60(6), 180(2).

As in previous cases x, y and z may become coefficients of growth; then m
as a determinant of the series becomes a constant complementary factor of both
variable co-factors of the series.

4. Generalization.

Let xj, xj, x 3 , .... x k _ 2 , x k _i, x k be the periodically alternating coefficients
of growth and m the determinant of a series.

(a) The order of T is kn + 1

Example:

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104

THEORY OF REGULARITY AND COORDINATION

/k-2\"

r x J x k n :i xr 1

|k{n-I)l + l

k

m x
1

k

rn | x

n-l -i

|k(n-2)] + I -

V 2
m | x

k^ r
m | x
1

|k(n-3)] + l

m | x

Tk (f)

= m (xJ

m

xf +1 [ x

Tk/n

1+2

= < m

*Odd number.
**Even number.

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UNIVERSITY OF MICHIGAN

CONTINUITY

105

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106

THEORY OF REGULARITY AND COORDINATION

k-3 1

T 3 =

n-1 n-1 n-1
x k-2 X k-1 X k

mX!X 2 X3

T 2 =

k-2\ n

1 /

mxix 2

T, = <m

Digil

■)"-]!

Google

mxj

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UNIVERSITY OF MICHIGAN

CONTINUITY

107

The limit indication in these formulae determines the products of all coeffi
cients of growth within the given limits.

Thus, X] = Xi

XjX2 = Xj X X2

k

x = X[ X X2 X X3 X Xjt

1

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108 THEORY OF REGULARITY AND COORDINATION

Figure 7. Rhythm is the law of regularity. A Schillmger design.

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UNIVERSITY OF MICHIGAN

CHAPTER 3
PERIODICITY*

HYTHM is the Law of Regularity in a factorial-fractional continuity. It is

the resultant of the synchronized component periodicities, their progres-
sion, modification and powers.

The simplest form of periodicity may be observed in longitudinal sine waves,
where the wave forms have definite regularity with respect to their frequency
and amplitude. Certain sound waves, moving through space, have such peri-
odicity.

A pendulum in its transverse oscillations 2 , adjusted to leave a trace, will
produce a wave form similar to the simplest wave. Such a wave motion expresses
uniformity in frequency, amplitude and phase, and is known in mechanics as
"simple harmonic motion".

The distance xi — x 2 forms a phase, and the distance Xi — x 3 forms a period.

Periodicity is a continuity of periods, or a continuity of phases. For the sake
of simplicity and graphic considerations we shall use the periodicity in phases as
units of measurement in further exposition. In order to establish values of
periodicities, we may neglect the amplitude of a phase (r). The graph of simple
harmonic motion, representing a progression of values along any parameter, may
be drawn in the following form.

'The reader is referred to Appendix A Basic 2 If a sheet of paper is placed under a pendulum

Forms of Regularity and Coordination for a detailed and moved in a direction perpendicular to its line

elaboration of the ideas and formulae presented in of motion, the pendulum will describe a sine

this chapter. See pages 445-639. (Ed.) curve. (Ed.)

Figure 1. Transverse oscillations of a pendulum.

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110

THEORY OF REGULARITY AND COORDINATION

I 1 I 1 I 1 ' 1

Figure 2. Simple harmonic motion graphed {sine curve).

The horizontal segments represent a progression of phases and the vertical seg-
ments define the limits between the adjacent phases.-

Single phases in a rectangular graph will be considered as standard units of
measurement.

We can define the periodicity representing "simple harmonic motion"
as monomial periodicity. The general form of monomial periodicity may be

expressed through consecutive terms: t|, t 2 , t 3 , which correspond to phases

on the graph; T is a definite periodic portion of continuity along any parametric
extension :

T = t, + t, + t, + t 4 + . ..t n (1)

The relative values of consecutive terms (which may correspond to absolute
values in any standard of measurement) may be expressed through a, h, c, ....
The expression for a definite portion of monomial periodic continuity will be:

T a = at, + at 4 + atj + . . .
T b = bt, + bt 2 + bt, + . . .
T c = ct, + ct 2 + ct, + . . .

T m = mti + mt 2 + mt 3 + . ..
By defining the limits "a" and "n" a more generalized form may be obtained.

n

T m = mti + mt 2 + mt 3 + + mt„ (2)

a

"m" may be equal to one inch, one second, one degree of an angle, one gram, etc.
These values, although arbitrarily chosen, will represent nevertheless an absolute
system of relations between the different periodicities.
Assuming that a = 2 and b = 3, then :

T a = 2ti + 2t 2 + 2t, + . . .
T b = 3t, + 3t 2 + 3t, + . . .

t a :t b = 2:3, *. the term of periodicity "a" is related to the term of peri-
odicity "b" as 2 to 3. From this progression ta can be expressed through t b :

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PERIODICITY

111

and t b can be expressed through t a :

t -- a
tb ~ 2

or, generally, if a = m and b = n,

t a :t b = m :n, then
mtb nt a

ta = — ; t b = — a

n m

A relation of two monomial periodicities through their absolute values may

1"

be represented graphically. If a = 1 and b = 2 in terms of — , then:

4

1" 1" 1"
T a = tl + t 2 + t, .... = -+-+-+ • • •

4 4 4

1" 1" 1"
T b =2t,+ 2t 2 +2t 3 ... = Y + 2~ + 2~ + "

Figure 3. Relation of two monomial periodicities.

This is the ratio of 1 :2, t. e., t a :t b = 1 :2. Thus, t a = ^ or t b = 2t a .

In terms of relative values (a = 1, b = 2) these periodicities may also be
expressed

T a = 1+1+1+1 + ....
T b = 2+ 2+ 2+ 2 + ...

1"

Assuming a = 1 = — , the graph will be enlarged accordingly.

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112 THEORY OF REGULARITY AND COORDINATION

Figure 4. Relative values of a and b increased.

Though the value of "a" has changed, the relation of "a" to "b" remains the
same.

A. Simultaneous Monomial Periodicities. (Synchronization)

/. Binomial relations of monomial periodicities.
(Synchronization of two periodicities)

The whole range of monomial periodicities, with respect to their relative
values and progression, may be represented through the series of natural fractions
previously described. We may write:

t
t

h + _ 2

2 2

tj U U

3 3 + 3

t1.t2.t3 t<
4+4 + 4 + 4

li + i? + h + + 5d

n n n n

ti , ti , tj , , t„ , 1 ^oo

00 ' 00 ' 00 ■ ' "'"oo"' '■' CO

The series from - to - bt T consists of an infinite number of series, where
1 00

T a = t, T b = 2t, T c = 3t, T d = 4t,
T n = nt and Too = 00 1.

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PERIODICITY

113

The derivative series resulting from the synchronization of T a and T b , where
a and b are fractions with any desirable denominator and a numerator of unity,
are obtainable through the interference of synchronized T a and T b . This process
will be realized as follows:

In order to express t a and t b in the same unities, it is necessary to find the
common denominator of a and b.

then

...)

...), then

T ' " 2 + 2

T ' ~ 3 + 3 + 3
The common denominator is 2 X 3 = 6.

Therefore :

Now t a and t b are equal: t a , b = - We may graph:

But, T a = at and
T b = bt,

T a = a(t t + ti + t, + . . .
T b = b(t, + t, + t, + . •

t a = - and
a

1

tb= b

The common denominator is ab.
Therefore, b is the complementary factor of a.
Assuming a = 2 and b = 3, we obtain

Figure 5. Synchronization of two monomial periodicities.

Original from
UNIVERSITY OF MICHIGAN

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114

THEORY OF REGULARITY AND COORDINATION

We now have two synchronized monomial periodicities, T 2 and Tj. The
derivative periodicity T 2sS resulting from the interference of phases, can be
obtained graphically by dropping perpendiculars from the beginning of each
phase of T a and T b .

Figure 6. Derivative periodicity of T it ».

Reading from the graph, we derive values for T a , b ,

_ 2t, t 2 t 3 2t«
Ta ' b ~ 6~ + 6 + 6 + 6~

Omitting the t's but preserving the same order of progression and writing
the numerators only, we discover the following rhythmic series:

T 2 „ = 2 + 1+1+2

3.

Any values may be selected for a and b. If - is a reducible fraction it should

b

be reduced first; otherwise the resulting rhythmic series will repeat itself. For
instance, % = -fa = etc.

We shall now assume a = 3 and b = 4.
Then:

T.-* + * + * + J

The common denominator is 3 X 4 = 12.

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PERIODICITY

Therefore,

t.jb = and we may graph:

4ti 4ti 4t_,

12 12 12

3ti 3t, 3ti 3t<

12 12 12 12

Tig r

Figure 7. Synchronization and derivative periodicity of T ttt .

The rhythmic series

T 3ti ^2 2tj 2tj I I ^5

lm ~ 12 + 12 + 12 + 12 + 12 + 12

Or, using the numerators only :

T„« = 3 + 1+ 2+ 2 + 1+ 3

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116 THEORY OF REGULARITY AND COORDINATION

Assuming various values for a and b, we get the following results:

Tj,8

2

+

2

+ 1

+

1

+

2

+ 2

Tst6

ST

3

+

2

+ 1

+

3

+

1

+ 2

+

3

T 4I j

—

4

+

1

+ 3

+

2

+

2

+ 3

+

1

+

4

TsiB

5

+

1

+ 4

+

2

+

3

+ 3

+

2

+

4

+ 1

+ 5

Tj,7

2

+

2

+ 2

+

1

+

1

+ 2

+

2

+

2

T3,7

=

3

+

3

+ 1

+

2

+

3

+ 2

+

1

+

3

+ 3

T 4l7

4

+

3

+ 1

+

4

+

2

+ 2

+

4

+

1

+ 3

+ 4

Tn7

5

+

2

+ 3

+

4

+

1

+ 5

+

1

+

4

+ 3

+ 2 +

5

Tg,7

6

+

1

+ 5

+

2

+

4

+ 3

+

3

+

4

+ 2

+ 5 +

1 +6

Tj,8

3

+

3

+ 2

+

1

+

3

+ 3

+

1

+

2

+ 3

+ 3

T 6 i8

5

+

3

+ 2

+

5

+

1

+ 4

+

4

+

1

+ 5

+ 2 +

3+5

T7I8

7

+

1

+ 6

+

2

+

5

+ 3

+

4

+

4

+ 3

+ 5 +

2+6

Rhythmic series resulting from the interference of two monomial periodic
groups a and b may be obtained by graphs or by computation. We shall consider
such a series as the resultant of interference, and we shall designate it as r(a-5-b).

The total number of terms in r(a-i-b) is:

N a:b =a+b-l (1)

The resultant always starts and ends with the term b, which also occurs in the
center of r, when

a + b — 1 = 2n + 1, i. e.,

when N a:b has an odd number of terms. We shall consider the b-terms, ap-
pearing at the beginning and at the end of r, as end-terms.
The number of b-terms in the entire r(a-r-b) is:

N b = a - b + 1 (2)

We shall consider - = m, where m is the integral part of the quotient,
b

The b-term appears at each end m times. Thus if m = 1 it appears once at the
beginning and once at the end of r; if m > 1, b-term, appearing m times at each
end of r, produces G b <, i. e., the end groups of b. The total number of b-groups
in the entire r is:

NG b = b (3)
The number of b-terms, appearing in the end-groups of r, is:

N b ,(G b .)=2m, (4)

as it is m at each end. Hence, the number of remaining b-terms is:

N b „ = N b - 2m = a - b + 1 - 2m (5)

It follows from the above, that the number of remaining groups of b-terms is:
NG b » = b - 2 (6)

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PERIODICITY

117

Hence the number of b"-terms, appearing in each G b -> in succession, is:

,^ , a — b + 1 — 2m
N b - (G b .) — , (7)

when G b = 2n, and

a - b + 1 - 2m

N b « (G b ,.) = g— ^ , (7a)

when G b = 2n •+■ 1-

The number of groups of x-terms appearing between the b-groups is:

NG X = a — b, when m = 1, (8)

and NG X = b — 1, when m > 1. (8a)

The number of x-terms in each Gx is:

2b - 2

N X (G X ) = when m = 1, (9)

a — b

2b - 2

and N X (G X ) = — — , when m > 1. (9a)

b — 1

When such a division is impossible without a remainder, two kinds of groups
develop between the b-groups: G x and G y . The number of all groups of G x and
G y remains the same, as if they all were G x , i.e.,

NG^ = NG„, i.e.: (8) and (8a).

The distribution of G x and G y can be obtained as follows. Through the
method of normal series, we can determine the possible places for the groups x
and y. Every resultant is a group with bifold symmetry. The geometric center
of the group is a point of symmetry. Thus, the second half of the series is the
inversion of the first half of it. Therefore, if the number of b-terms does not
exceed 3 when a < 2b, or 5 when a > 2b, the two groups being on symmetrical
places can be x only.

b...x...b...x...b
or b + b...x...b...x...b+b
With 4b when a < 2b and 6b when a > 2b, there is only one solution for the
places of groups between the b's.

b...x...b...y...b...x...b
or b-f b...x...b...y...b...x...b+b
With 5b when a < 2b and 7b when a > 2b, one of the b's coincides with the
point of symmetry. Thus, we obtain x and y on each side of the series.

b...x...b...y...b...y...b...x...b
b + b...x...b...y...b...y...b...x...B+b
Some other examples:

b appears 6 times —
b. ..x. ..b. . . y . ..b. ..x. ..b. . . y . . .b. ..x. ..b
b...x...b...x...b...y...b...x...b...x...b

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118 THEORY OF REGULARITY AND COORDINATION

b appears 7 times —
b. . .x. ..b. . .y. ..b. ..x. ..b. ..x. ..b. . . y . ..b. ..x. ..b
b...x...b...y...b...y...b...y...b...y...b...x...b
b. ..x. ..b. ..x. ..b. . .y. . .b. ..y. . .b. ..x. ..b. ..x. ..b

As 2 has always two identical conversely arranged halves, it is necessary to
find individual terms only for -. The values of terms x and y in the groups of

x and y are controlled by two basic conditions:

(1) the values are represented by continuously alternating binomials:
a — b and b — a;

(2) the coefficients of these binomials express the following regularity:

(a) cases, where a — b > 2 :

coefficients of a: 1, 1, 2, 2, 3, 3, ... m, m;
coefficients of b: 1, 1, 2, 2, 3, 3, ... m, m;

(b) cases, where a — b > 2 :

coefficients of a: 1, 1, 2, 2, 3, 3, ... m, m;
coefficients of b: 1, 2, 3, 4, 5, 6, m.

Examples:

(1) r(7 + 6) - b + (a - b) + (2b - a) + (2a - 2b) + (3b - 2a) + . . .

(2) r(9v5)=b + (a-b) + (2b - a) + (3b - 2b) + (2a - 3b) +

+ (4b - 2a) + . . .
where 3b — 2b = b.

2. Synchronization of two monomial periodicities with consecutive displace-
ments of periodicity b with regard to the integral phases of periodicity a.

Through this process the resulting periodicity becomes fractioned on both sides
of the point of symmetry. When b = n, n single units will appear in succession
in the center of the resulting periodicity. In order to obtain such a result, it is
necessary to use b as a factor for the number of terms in both a and b period-
icities.

Thus, T a — — + — +.. . b times
a a

or T,

This periodicity being in synchronization with periodicity b will appear only
once.

Also, T b = + ~ + . • b times
b b

or

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PERIODICITY

119

Periodicity T b in its consecutive displacements towards the phases of periodicity
T. in synchronization will appear a — b + 1 times.

NT, = 1

NT h = a - b + 1
Now we can represent the whole scheme of such synchronization:

Ta ^ + L* + t -' + t < + ...
a a a a

^ b + b + b + b •

Ta 2

Ta 3

ti t 2 t 3 U
-+-+-+-+
b b b b

+ + +
b b b b

Assuming values for a and b

a = 3 ; b = 4, we can represent this process graphically.

NT b =4-3 + 1 =2

As in previous cases of synchronization we find the common denominator first

t _ 1 t 1 J_
a~3 : b"4 t, + b ~12

The complementary factor for T a is b and for b the factor is a. Therefore, the
unit of periodicity a in synchronization is

T b _ J
*~ab~12

T _a _ J
b ~ab ~ 12

b being a factor for both periodicities in synchronization will give

* ab ab ab ab

X = ^ + ^ + ^4.!!?
' ab ab ab ab

or

T.-*(£)andT ( -4(l)

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120

THEORY OF REGULARITY AND COORDINATION

£ r

? r

ft.

J

3t M

3t±

3*.
\ I

3t 9 3t*

J 1 I

Figure 8. Consecutive displacement of periodicity b.

Through interference, we may obtain the desired resulting series with fractioning
around the point of symmetry.

Figure 9. Derivative periodicity.

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PERIODICITY

121

_ 3t, t s 2t, t 6 t« t 7

Ts:(2X4) ~ 112 + 12 + 12 + 12 + 12 + 12 + 12 +

2U , t, 3t l0 .
H — - H — - + 7T , or using numerators only,
12 12 12

T, :( 2x«)= 3 + 1+ 2 + 1+1+1+1+2 + 1+ 3
Let us now assume that a = 2 and b = 5. Then,

= 2 ' tb = 5 ' therefore ta:b = Jo
In synchronization, their values will be:

b 5 a_ 2_

l= ab"lO : b= ab~10
b being a factor for both periodicities in synchronization will give:

x = ^ + ^ + ^ + ^ + ^
* ab ab ab ab ab

X _ tia | tta | t,a } t<a \ tia
b ab ab ab ab ab

or

-,- 5 (i)a„dT.= 5 (i)

NT b = 5—2 + 1
Now we can synchronize T a with T b

Figure 10. Synchronisation and derivative periodicity of 7\,» with consecutive

displacement of b.

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122

THEORY OF REGULARITY AND COORDINATION

Tj,(4x5) =

+

2t,

+

2t,

+ t.

+

t«

+

+

ts

+

t 7

+

i« + i- +

25

25

25

25

25

25

25

25 25

tu

+

tlS

+ t "

+

tu

+

tie

+

tl7

+

tu

+

tu , 2t»o

25

25

25

25

25

25

25

25 "2T

tip 1 tll 1

25 25

+

2t

21

25

or, using numerators only

T, ,(«»., = 2 + 2 + 1+1+1+1+1+1+1+1+1+1+1+1.
+1+1+1+1+1+2+2

Assuming various values for a and b we get the following results:

T„<,«« =2 + 1+1+1+1

T„ (2 x4) = 3 + 1+ 2 + 1+1+1+1+2 + 1+ 3

T„ ( ,«,> = 3+ 2 + 1+ 2 + 1+1 + 1+1+1+1+1 + 1+1+2 +
+1+2+3

T 4 , (1 ,o = 4 + 1+ 3 + 1+1+2 + 1+ 2 + 1+1+3 + 1+ 4
T 4 , ( 4.7) = 4+ 3 + 1 + 3+1+2 + 1+1+2 + 1+1+1 + 1 + 1 + 1 +

+ 1+1+1 + 1+1 + 1 + 1 + 2 + 1 + 1 + 2 + 1 + 3+1 + 3 + 4
T„ ( ,,fl = 5 + 1+ 4 + 1+1+3 + 1+ 2+ 2 + 1+ 3 + 1+1+1
T il(J x7) = 5 + 2+ 3 + 2+ 2 + 1+ 2+ 2 + 1+1+1+2 + 1+ 2 +

+ 1+1+1+2+2 + 1+ 2+ 2+ 3+ 2+ 5

After synchronization of both periodicities, a being a major term and b being a
minor term, we may obtain the whole number of terms in the resulting periodicity
through the following formula:

NT = a (a - b + 1) + (b - 1) = a J - ab + a + b - 1 (1)

3. Balance.

Formulae :

(1) When a > 2b, a > 3b, a > 4b a > mb

B» >b = r. + b + r. + b + a (a - b)

(2) When a > 2b, a > 3b, a > 4b, a > mb

B. >2b = r^ + 2r a + „ + (a 2 - 2ab)
B a >3b = r, + b + 3r. + „ + (a 1 - 3ab)
B.> 4b = r. + „ + 4r, + b + (a 2 - 4ab)

B a>mb = r 4 + b + mr, + b + (a 2 - mab)

GENERAL FORMULA

Bomb = r, + b + mr. + b + (a* - mab)

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123

4. Application

5 4-2

-

(1)

2 + r 5 + 2 + 5 (5

- 2)

= 25t

4- 10t + ISt

(2)

B B + 2 = r& +

2 + 2r 6 + 2 + (25 -

- 20)

= 25t

+ 10t + 10t + 5t

7 -r 3

(1)

B 7 + 3 = r 7 +

, + r 7 + , + 7 (7

-3)

= 49t

+ 21t + 28t

(2)

Bt + 3 = O +

, + 2r 7 + , + (49

-42)

= 49t

+ 21t + 21t 4- 7t

8-^3

(1)

Bg _,. 3 = Tg +

3 "1" rg + 3 + 8 (8

- 3)

= 64t

+ 24t + 40t

(2)

Bg + 3 = rg +

s + 2r 8 + j + (64

- 48)

= 64t

4- 24t + 24t 4- 16t

9 v4

(1)

B» +. 4 = r» +

j + r, + 4 + 9 (9

-4)

= 81 1

+ 36t + 45t

(2)

B9 + 4 = r* +

_4 + 2r 9 + 4 + (81

- 72)

= 81t

4- 36t 4- 36t + 9t

7-^2

(1)

B7 + 2 = O +

2 + r 7 + 2 + 7 (7

- 2)

= 49t

4- 14t + 35t

(2)

B7 + 2 = r 7 +

2 + 3r, + 2 + (49

-42)

= 49t

4- 14t + 14t 4- 14t + 7t

9-^2

(1)

Bg + 2 = T<t +

2 + r 9 + 2 + 9 (9

- 2)

= 81t

4- 18t + 63t

(2)

B» ^ 2 = r» +

"2 + 4r, + 2 + (81

- 72)

= 81t

4- 18t + 18t 4- 18t + 18t4-9t

B. Polynomial Relations of Monomial Periodicities.

(Synchronization of several periodicities).

As in the two previous cases, the resultant rhythmic series derives from the
interference of monomial periodicities, synchronized through a common denomi-
nator. If we have three monomial periodicities T„ T b and T c , they should be

synchronized through T a . b . c where each unit in synchronization is— — .

abc

The complementary factor for a is be.
The complementary factor for b is ac.
The complementary factor for c is ab.

Each unit of the synchronized continuity will take the following form:

_ /bct\ bcti . bct 2 . , i . ,
T a = a ( — - J = — — H — r— 4- ••• [a times]
\abc/ abc abc

_ /act\ acti . act 2 , „. ,

T b = b ( — J = -— 4- — - + . .. [b times]
\abc/ abc abc

_ /abt\ abti . abt2 , r ^. ,
T c = c ( — ) = —- 4- —- 1 + . . [c times]
\abc / abc abc

rv -■ f\r\cs\{> Original from

Digitized by ^WJjl^ UNIVERSITY OF MICHIGAN

124

THEORY OF REGULARITY AND COORDINATION

Assuming a = 2, b = 3 and c = S, we obtain
Tl 2" + 2

t ' = j + j + j

The common denominator is 2 X 3 X 5 =30

Therefore T, - 2 (^) - J£ + ^
\ 30 / 30 30

/I0t\_10t, lOt,
Tl HjO/ 1 - 30 + 30 +

10t,
30

_ _ / 6t\ 6t, 6t 2 6tj 6t 4 , dt s

1 1 = 5 I — J = — i ~r -

\30/ 30 30 30 30 30

Thus, ti, J,* = —
Now we may graph :

' I ^ 9 f

3i:

i

Figure 11. Synchronisation of three monomial periodicities.

The resulting rhythmic series:
T

30

6ti , 4t, + 2ti 3t, , 3t s , 2_U , . 6t,
30 30 30 30 30 30 30

Google

Original from
UNIVERSITY OF MICHIGAN

PERIODICITY 125

or using numerators only:

T SlllB = 6+4 + 2+ 3+ 3+ 2+ 4+ 6
Another example of three synchronized monomial periodicities.

T4, i±i

a

1

■■

y

-

F

-

3

\

:

"3

j

u

L

1

Figure 12. Synchronization of Tai*^.

- r , ... . . T 12ti 3t, 5t 4 4t 4 6t* 6t«

I he resulting rhythmic scries 1 jius— r — T — + 1 t — *r

* * 60 60 60 60 60 60

, 4t 7 . St. 3t s . 12t 10

i H

60 60 60 60

Or, using numerators only:

Ti.ul- 12+3+5+4 + 6+6 + 4 + 5+ 3 + 12

Likewise:

T„ 417 = 12+9+3+4 + 8+ 6 + 6 + 8+ 4 + 3 + 9+12

In some cases of the synchronization of several monomial periodicities, the
component periodicities neutralize each other through interference, because of
their numerical relations. For instance, in the case

T SlJlT = 7+7+7+7+7 + 7,

results in a monomial periodicity.

In order to obtain a resulting rhythmic series from four or more monomial
periodicities synchronized, it is necessary to follow the procedure previously
described.

rv -■ _j l r\rscs\f> Original from

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■

126

THEORY OF REGULARITY AND COORDINATION

Here is the process for synchronizing monomial periodicities in more gen-
eralized form:

(1) Select the number of component monomial periodicities to be synchron-
ized. For instance, suppose we take n periodicities.

(2) Assume certain values (in integer terms) for all the component perio-
dicities. For instance,

n = a, b, c m

(3) The values a, b, c, ... will not contain a common divisor. Find the
common denominator for the periodicities to be synchronized. It is a product of
a by b, by c, etc. If T a represents the periodicity which will appear a times in
synchronization, an analogous interpretation will be given to

T„, T c , ... etc.

Then the common denominator or the unit of periodicities T,, T b , T c , . . . .T m
in synchronization can be expressed through

T a: b:c:. . . :m = abc ...III Or t = —

abc. . . m

(4) Find the complementary factor for each component periodicity in
synchronization. The complementary factor equals the product of the remaining
terms. Thus, for

T» the complementary factor equals

. abc . . . m
be ... m or

a

T b the complementary factor equals

abc . . . m
ac . . . m or

b

T c the complementary factor equals

, abc . . . m
ab . . . m or — -

T m the complementary factor equals

. abc . . . m

abc ... e or

e

(5) Now the component periodicities can be expressed in common units
and synchronized

^abed .

. . m,

'acd . .

. mt

v,abcd .

. . m

abd . .

.me

)

m \abcd . . . m/

rv -■ _j l r\rscs\f> Original from

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PERIODICITY

127

Let us assume n = 5, where a = 2, b = 3, c = 5, d = 7, e = 11

Then,T 2:3:5;7:1I = 2X3X5X7X11 = 2310

1_

1 " 2310

The complementary factor for
2310

T 2 = — = 1155

_ 2310

T, = — - = 770

2310 „„
T. = ~-462

T, = = 330

7

2310 „
T„= - — = 210
11

Now we may synchronize

1155t\ 1 155ti 1155t»
2310 2310

T = 3 ( 770t \ = ]™h + IZPi! + 770ti
3 \2310/ 2310 2310 2310

T _ 5 - 462t i + *62t 2 462t 3 462t« 462t &

& ~ \2310/ ~ 2310 2310 2310 2310 2310

_ 7 / 330t \ _ 330ti 330ti 330t 7
7 ~ \2310/ ~ 2310 2310 2310

• _ u( 2m \ =^i + 2 i^l J + + 210tn

11 \2310/ 2310 2310 2310

The general number of terms in any resultant rhythmic series, produced
through synchronization of n monomial periodicities, equals the sum of all the
component periodicities, minus n — 1.

GNT a:b:c: . ... :m = (a + b +c + ... m) - (n - 1) =
a + b+ c+ ... + m_ n + l

In order to produce more extended rhythmic series, where the number of terms
will be considerably greater, while their value is considerably smaller, it is neces-
sary to synchronize the component monomial periodicities through the common
product, instead of the common denominator. This can be performed with three
or more component periodicities.

If T a , T b and T c are the component periodicities, their common product

will be T a:b:c = abc and t = -7— .

abc

Digitized byGoOgle

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UNIVERSITY OF MICHIGAN

128 THEORY OF REGULARITY AND COORDINATION

Then, T a = be (at) = ati + ati + .... [be times)
T„ = ac (bt) = bti + btj + .... [ac times]
T c - ab (ct) = ctj + ctj + .... jab times]

Assuming a = 5, b = 3, c = 2, we get

1

T t:3: i= 30 and t = — -

30

T, = 6(5t) - 5ti + 5t, + 5ta + 5t* + 5t» + 5t,
T, = 10(3t) = 3t» + 3ti + 3t, + ■ • • + 3t M
Ti = 15(2t) " 2t, + 2U + 2t, + . . . + 2tu

Now we can graph

-

-

■

-

?

7

-

■

-*

-

I

>-

-

f

T

1

1

E

f

(

I

i

1

'■

Figure 13. Synchronization of rj :3:! .

Thus, T, :I:S = 2t, + t, + t, + U + U + 2t» + U + t 8 + 2t, + 2t M +

+ t„ + tu + 2t I3 + 2t, 4 + t u + t„ + 2t 17 + tit + tn +
+ t» + tsi + 2t !2

Or, simply, T B:J:I = 2+1+1+1+1+2+1+1+2+2+1+1+
+2+2+1+1+2+1+1+1+1+2

Here are some other rhythmic series obtained through the common product of
three periodicities:

T 6: 4 :a = 3 + 1+1+1+2 + 1+1+2 + 3 + 1+ 2 + 2 + 1+ 3 +
+ 1+ 2 + 1+ 2+ 2 + 1+ 2 + 1+ 3 + 1+ 2 + 2 + 1+3 +
+2+1+1+2+1+1+1+3

T,. 1: , = 2 + 1 + 1+ 2 + 1+1+1+1+2+2 + 1+1 + 2 + 2 +
+ 1+1+2+2 + 1+1 + 2 + 2 + 1 +1+1 +1+2 + 1 +
+ 1+2

Digitized byGoOgle

Original from
UNIVERSITY OF MICHIGAN

PERIODICITY

129

T 7:4 : t = 2+ 2 + 1+1+1+1-1-2+2+2 + 1+1+2 + 2 + 1 +

+ 1+ 2 + 1+1+2+2+2 + 2 + 1+1+2+2 + 2 + 2 +
+ 1+1+2 + 1+1+2 + 2 + 1+1+2+2+2+1 + 1 +
+1+1+2+2

T 7 :4:» = 3 + 1+ 2 + 1+1+1+3+2 + 1+1+2+2 + 1 + 3 +

+ 3 + 1 + 2+ 2 + 1 + 2 + 1 + 3 + 1 + 2+ 2+ 2+1+3 +

+ 1+ 2 + 1+ 2+ 2 + 1+ 3+ 3 + 1+ 2+ 2 +1 + 1 + 2 +
+3+1+1+1+2+1+3

T 7:i:4 = 4 + 1+ 2 + 1+ 2+ 2+ 2 + 1+1+4 + 1+ 3 + 1+ 3 +

+ 2+ 2+ 3 + 1+ 4 + 2+ 2 + 1+ 3 + 1+1+2+3 + 1 +

+ 4+ 3 + 1+1+3+2+2+3 + 1+1+3+4 + 1+ 3 +

+ 2 + 1 + 1+ 3 + 1+ 2 + 2 + 4 + 1+ 3+ 2+ 2+ 3 + 1 +
+3+1+4+1+1+2+2+2+1+2+1+4

Tt:*, = 3 + 2 + 1 + 1 + 2 + 1+ 2 + 2 + 1+ 3+ 2 + 1+ 3 + 1 +
+2+1+2+3+2+1+3+1+2+3+3+1+1+1+
+3+1+1+1+3+3+2+1+3+1+2+3+2+1+
+2+1+3+1+2+3+1+2+2+1+2+1+1+2+
+ 3

C. Polynomial Relations of Polynomial Periodic Series.
(Synchronization of various periodic series).
Once the method of synchronizing through common denominator or product
has been established, all other cases of polynomial relations of various periodic
series can be classified, and the resulting series deduced.

(1) If T a+b and T',<+b' are equal in the sum of their binomial as well as
in their corresponding terms (a = a, b = b), the resulting periodic series will
be equal to any of the two component series T. In this case, the results valuable
for art purposes may be obtained by applying various coefficients to T and T' and
by synchronizing the latter.

For example, if we assume

then, T = m(a + b) = (a + b)t, + (a + b)t! + . . . + (a + b)tm

T= n(a' +b') = (a' + b')U + (a' + b')t, + . . . + (a' + b')t n

/. Binomial relations of binomial periodic series.

T,

Now we can synchronize.

„, /na + nb\
T m - m ( J t =

\ mn /

(na + nb\ t , / na + nb
mn / V mn

t, + . . . . +

Digii

Google

Original from
UNIVERSITY OF MICHIGAN

130

THEORY OF REGULARITY AND COORDINATION

/ ma + mb\ / ma + mb\ , / ma + mb\

.„= nl ) t = I J tt + l 1

\ mn / \ mn / \ mn /

V mn /

Let us assume that a = 2, b = I, and m = 4, n = 3.

r., (3X2)+(3X1) . , (3X2) + (3X1) ^ , <3X2)+(3Xl) ^ ,
Then, T« = 12 t, + ^ *« + y *-+

, (3X2) + (3X1) , 6+3 ,6+3 ,6+3 ,6+3

+ n — ''-Ij- tl +lF t,+ lf t>+ lT l *

_ (4X2)+(4X1) , , (4X2)+C4X1) . (4X2)+(4X1) ^
Tl = Vl tl + 12 ti + 12 t> =

8X4 8X4 8X4

"l2" tl+ l2" tt+ ~n u

or, graphically
scale
t. « 4

t b = 3

The whole scheme of synchronization will take
• *

Tra:n<m+u) = mn(a+b)
T«, a(l Kij = 12 X 3 = 36

E;ii:i5:*

It

4*

Digiti

Figure 14. 4:3 synchronisation of 2 + /.

CrkOolf> Original from

V,UU S UNIVERSITY OF MICHIGAN

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PERIODICITY

131

The resulting series is

t _ 6tl -i- 2 ±1 -u h 4- 3t « _i_ 3ts _u 3t « 4. 2t7 _l 4t « _i_ 3t » >

T M M - j 6 + 36 + 36 + 36 + 36 + 36 + 36 + 36 + 36 +

5tio _j_ tn _|_ 3tu
^36 36 ^6

or, 6+2 + 1+3+3+3+2+4+3+5 + 1+3

(2) a a' and b ^ b', but, a+b=a'+b'=s
In this case the whole scheme of synchronization can be performed directly from
s in 1 : 1 = s : s synchronization.

Let us assume that a = 5, b = 3

a' = 6, b' = 2

Then

r I l— j

T* \ n i

I ill

i iii

i i i i

i i i

7? 7" | LJ 1

Figure 15. a a' and b ^ b'.

W^^^ + ^ + ^orS + l +2

A special case when a = b' and b = a' is typical in art. It yields a synchroni-
zation of a binomial and its converse.

a - 3, b - 1

a'-l,b'-3

Digitized byGoOgle

Original from
UNIVERSITY OF MICHIGAN

132

THEORY OF REGULARITY AND COORDINATION

T
T'

t.t'

Figure 16. a — b' and b = a'.

— ti . 2tt . ti * , * , *

T ( , +1)!{1 + „ = - + — + - or l+2 + l

Synchronization may also be performed through the coefficients independent of
the value of 8 or its component terms. This will be realized through a common
product.

For example, 4 + 1= 5

1+4=5 s — 5, m — 3, n— 2

1- 3(1 +4) ] 3+12 3 + 12

but T m:n(i) - mns

T| SfW - 3X2X5 - 30
1

'"56

or graphically :

r\r\Ci\f> Original from

Digitized by ^UUglL UNIVERSITY OF MICHIGAN

PERIODICITY 133

Figure 17, S M+b = 5V +b <.

3ti Stj , 2U , 5t« , 3t t , 2t, , 8t T , 2t,
T,:1W " B + 3^ + 35 + B + 30 + 30 + 30 + 30

or, 3+5+2+5+3+2+8+2
(3)

When the sum of one binomial does not equal the sum of another, S and S'
act as complementary factors in synchronization.

_ s/— ^ - — + — -I +

' \SS7 ~ SS' SS' " ' SS'

T-s/^ = ^ + ^ + + — '

Vssv SS' SS' SS'

Let us assume that S = 3 + 2 = 5
and S'~ 4 + 3 - 7

T| - , + 1-7 U/ 35 + 35

+ ... +

T 7 _i+i

Digiti;

V35/ 35 35

• 00

'8 lc

5tr
35

35

Original from
UNIVERSITY OF MICHIGAN

134

THEORY OF REGULARITY AND COORDINATION

Now we may graph

-

.

H

-1

H

f+

■f

M

*-

-

.

■

/

.'

I

V

! .

.

j

L

-

[

3ti , ti . ti . 2ti . tt , 2t« , ., , , . »

T l(l+1)!l( . + 1) - ~ + - + -+ -+ - + -+ .._ + „ -r ,

_j_ _|_ _j_ -I- -j_ ^* 14 _1_ _L ^1* _(_ ^' 1T _|_

35 ¥ 35 Is" 35~ "35~ 35~

or.

3 + 1+1+2+1+2+1+2 + 1 + 1+3+2 + 1+ 2+2+3 + 2+2+1+2

35

+

35

+

35

2t 1B

+

tis

+

2tu

35

35

35

2. Synchronisation of a Motive.
2 + 1+1

inl*2+3-i-4
Common product = 12 12 (2 + 1+1) 1

6(2 + 1+1) 2

i-J>

Digitized by GoOgle

4(2 + 1+1) 3

3 (2 + 1 + t) 4

Original from
UNIVERSITY OF MICHIGAN

PERIODICITY

135

k» *

mm

-0

-0

^ 1

H

» —

. ^-ln -

- -

o_ —

-= —

- J Yr

Figure 19. Synchronization of a musical motive 2 + 1 + 1.

Original from

Digil

.oogle

UNIVERSITY OF MICHIGAN

136 THEORY OF REGULARITY AND COORDINATION

*

3. Polynomial Relations of Binomial Periodic Series.

This case involves the synchronization of several binomial periodic series.
Let us assume that

A = ai + aj
B = bi + bi

C = cj -f ct .

M = mi + mi,

where A, B, C, . . . .M are the sums of the corresponding binomials ai -f a J(
bi + bj, Ci + cj, . . . mi + mj. The sums of the component binomial periodicities
enter as factors of the common product ABC. . . .M. The complementary factor
for each component periodicity equals the general product divided by a given
periodicity. The complementary factor for

A equals BC. . . .M
B " AC....M
C " AB....M

M " ABC...

Now the component binomial periodicities can be expressed in common units
and synchronized.

T A -», + a,-BC...M( At \. (»» + »'>'» + («' + «'>'» +,..

VABC...M/ ABC. .M ABC...M

, (ai + ai)tBc ,m
ABC...M

T B =b 1 + b I = AC u(— * (b i ±b ! )l» + (b, + b,)t.

\ABC. . . M/ ABC. . .M ABC. . .M

, (b i + bj)t AC , m
ABC...M

Tc = c t + ci = AB . . . M (— g- 1 -) = (C1 + C ° U + (C1 + Ca) U + . . .

\ABC. . . M/ ABC . .M ABC M

, (Cl + C t ) t AB M

ABC M

T M =m 1+ m 2 = ABC. -) = ^1+"^ + ("»»+"")t» + . . .

\ABC. . . M/ ABC M ABC . M

■ (mi + m t )t AB c
ABC M

Digitized by GoOgle

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UNIVERSITY OF MICHIGAN

PERIODICITY

137

Example:

Let us assume that 3=2 + 1,5=3+2, 7=4+3, 8=5+3

T 3 :5 7:8 = 3X5X7X8 = 840

_ J_
* ~ 840

The complementary factors are :

h " So " 280
u " So " 168

7

( ' " So " 120
'•-So' 105

Now we can write:

Tj = = (2 + m, (2 + l)t„

\840/ 840 840 840

x ,«/ St \ (3 + 2)t, , (3 + 2)t, , , (3+2)ti»

T6 = 168 W = -8l0— +_ 840~ + + 840

T7 = 1M (JL) _ (4 + 3)t, (4 + 3)f (4 + 3) tM .

\840/ 840 840 " 840

x lft c/ 8t \ (5 + 3)t, , (5 + 3)t 2 , , (5 + 3)t 1Bt

Ts = 105 W = — 8lb~" + "lib" + ■ + 840

The graph may be made according to the method previously used.

4. Binomial relations of polynomial periodic series.
Let us assume two polynomial periodic series:
S=a + b+ c+ d and
S' = a' + b' + c' + d' + e' where

(1) all of the terms are respectively equal — a = a', b = b\ c = c', d = d',
while the number of terms in S and S' may vary;

(2) some of the terms are respectively equal — a = a', or d = d', or irrespec-
tively equal a = c', or d = b';

rv -■ _j l r\rscs\f> Original from

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138 THEORY OF REGULARITY AND COORDINATION

(3) none of the terms of the two series is equal. As in case 2 above,

T S:S - = SXS'=SS'; t = ^

-'(§) =

(a + b + c + d)t, (a + b + c + d)t.

+

SS' SS'
(a + b + c + d)tv

T s ,=

SS'

s /S^t \ = (a+b+c + d + e)t, (a-f b+c+d+e)t,
\SSV SS' SS'

(a+b+c+d +e)t s

+

SS'

In the case of the two polynomial periodic series, the number of terms in each
series may be the same or different.

S = m, S' = m or S = m, S' = n

Example:

S = 5= 2+ 2 + l (3 terms)

S' = 7= 3+ 2 + 1+1 (4 terms)

T S:S ' = 5X7 =35; t = ~

•-(B)-
■ 5 (B) -

(2 + 2 + l)U (2 + 2 + l)t, (2 + 2 + l)t 7

35 35 " * " 35

(3 + 2 + 1 + l)t, (3 + 2 + 1 + l)t,

. .. +

35 35

(3 + 2 + 1 + l)t s
35

5. Polynomial relations of polynomial periodic series.

As in Case 3 above, some of the terms of one periodic series may or may not
be equal to the terms of another series. The process of synchronization develops
as in Case 2. Let us assume m polynomial periodic series with a variable number
of terms.

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PERIODICITY

139

A = ai + a 2 (2 terms)

B = bi + b, + b, + b 4 + b 5 (5 terms)

C = ci + c 2 + c s (3 terms)

M = mi + mj + mj (3 terms)

1

T A;B :C: . M = ABC. . .M; t
Now we can synchronize.

ABC. . .M

T _ rc M ( At ^ _ (ai + a?)tl I (ai + 3t)tl I
A \ABC...M/ ABC...M ABC...M

(ai + ai)tBC...M

.. +

ABC . . .M

/ Bt \ = (bi+b,+b,+b«+b t )ti
Tb_AC M VABC...M/~ ABC...M

(b 1 +b > +b > 4-b 4 +b t )ti , (bi+^+bj+bj+b^tAc. M

+ ABC...M + •• + ABC...M

/ Ct \ _ (ci + c» + ci)ti (ci+c« + ca)t»
Tc_AB M \ABC...M/ ABC...M + ABC...M

(Ci+Cj+C|)t AB M

+

ABC...M

/ Mt \ _ (mi -I- m a + m s )U
T M = ABC. \ ABC M ) ~ ABC. . .M

(mi + m 2 + m 3 )t 2 (m t + m 2 + m»)t ABC

+ ABC...M + •■ + ABC ...M

Example:

A = 4 = 2 + 1 + 1
B = 5 = 3 + 2
C=7=3+l+2+l

T 4:S :7 =4X5X7 = 140; t = —

_ /jt\ _ (2 + 1 + Dtt

- 35 ^ 140 ; - 140 +

(2 + 1 + l)t„

+ l)ti , (2 + 1 + l)t»
140

+ 140

rv -■ _j l r\rscs\f> Original from

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140) THEORY OF REGULARITY AND COORDINATION

x 98 ( 5t ^ ( 3 ± 2 )t' , O + 2)t, ^ (3 + 2)t„

x on/ 7 ^ (3 + l+ 2+l)t, , (3 + l+2 + l)t, ,

(3 + 1 + 2 + l)t„
140

Practical Application in Art.

For practical application in art, any binomial or polynomial periodic series
may be chosen from the rhythmic series resulting through interference. AH the
rhythmic series obtained in such a way have two fundamental characteristics:

(1) inverted symmetry

(2) maximum of balance at the point of symmetry and minimum of
balance at the starting and ending terms.

In the series with fractioning around the point of symmetry, the balance is pro-
duced by a group of uniform units. In the series with an odd number of terms,
the most balanced binomial usually falls at the beginning and at the end of the
series. For instance:

T 6: , = 3+ 2 + 1-1-3 + 1+ 2+ 3
I 1 I 1

In this case 3 + 2 is the most balanced binomial with S = 5.

For expressive, contrasting, dramatic effects it is more appropriate to use
the unbalanced binomials. For calm, inexpressive, persistent, monotonous or
contemplative effects, it is more appropriate to use the balanced binomial or
polynomial groups. Using both these types of groups, unbalanced and balanced,
produces an effect of relaxation and the relief of tension. The progression of
balanced and unbalanced groups may also be subjected to interference, as two
elements U (unbalanced) and B (balanced).

All resulting rhythmic series may be used in their entirety, or in halves (from
the beginning to the middle term or to the point of symmetry), or in selected
segments. A rhythmic series (or a part of it) once selected becomes a thematic
entity, and often the foundation not only of one individual composition, but of
the whole style of an epoch or a nation. We shall analyze this more thoroughly
in the succeeding chapters, and refer now to a few typical cases.

European music of the 17th to 19th centuries operates mainly on binomials,
2 + 1 and 3 + 1, and a trinomial, 2 + 1 +1, with permutations. In its original
idiom, Russian folk music employs 3 + 2 and 2 + 3, 2 + 3 + 2 and 3+2+3.
Today's American jazz has a Charleston foundation, probably imported from the

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PERIODICITY

141

Caribbean, where it is most common in folk music (compare the dance songs of
Puerto Rico). Its binomial is 5 + 3 and 3 + 5; its trinomial is further fractioning
of the same binomial, 3+2+3, with permutations.*

To fit any particular style, one may select the corresponding balanced or
unbalanced groups from the appropriate rhythmic series. For instance, if one
desires to operate on Charleston rhythms, in other words, on a binomial S = 8,
we can produce this binomial from the whole series of possible binomials within
the sum 8:

The most balanced binomial in this case is 5 + 3 or 3 + 5 (4 + 4 is excluded as
reducible to 1 + 1). By selecting these binomials from the appropriate rhythmic
series, we gain because we find all the variation of a selected idiom and in a
properly arranged progression, ready to use. If S = 8 and a + b = 5 + 3, all the
series resulting from the interference of 8:5, 8:3, 8:5:3 will satisfy the style.
In this respect the general formula of style ratios may be expressed:

S=a+b+c + ... + m
S : a ; S:b; S:c;....S:m
S : a : b; S : a : c; . . . .S: a : m

S:b:c; S:b:d; S:b:m

S:a:b:c; S: a:b:d;....S:a:b:m

S : a : b : c : : m

Here S, a, b, c, . . . . m are the style determinants.

E. Synchronization of the Second Order.

A polynomial rhythmic series can be synchronized with itself or with any
other rhythmic series under various coefficients, and as many times as desired.

1. Synchronization of a rhythmic series with itself through various coefficients:
m, n, o, p, q,

Suppose we have a rhythmic series

7 + 1, 6 + 2, 5 + 3, 4 + 4, 3 + 5, 2 + 6, 1 + 7.

T .

1 x:y

=a+b+c+c+b+a

Let us synchronize it through m, n, p.

Then

T m:ii: p( x:y , = mnp (x:y);

1

t =

mnp (x:y)

*These quantities refer mainly to rhythmic du-
rations. (Ed.)

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142 THEORY OF REGULARITY AND COORDINATION

[~m(x:v)t1 rm(a+b+c+c+b+a)tl

T m (x:y) = np = np =

L mnp J L mnp J

[(ma + mb + rnc + mc + mb + ma)t~|
=
mnp J

mnp

(ma + mb + mc + mc + mb + ma)ti
mnp

+

^ (ma + mb -f- mc + mc + mb + ma)t 2 ^
mnp

( ma + mb + mc + mc + mb ~f- ma)t np
mnp

rn(x:y)t"l [n(a + b + c +c + b + a)t~]

T n( x:y) = Hip = m P — — =

L mnp J L mnp J

t (na + nb + nc + nc + nb + na)t "| _
mnp J

mnp J L mnp

T(n;

= mp

L ITlIip
4- nh 4- n r 4- pc 4- nh 4- n;i It > ^

mnp

(na + nb + nc + nc + n b + na)ti
mnp

(na + nb + nc + nc + nb + na )U ^
mnp

(na + nb + nc + nc + nb + na)t mp
mnp

r P (x:y)t1 fp( a + b + c + c + b + a ) t l

T »<-> = h^~J = mn L n^p" J "

|~(pa + pb + pc + pc + pb + pa)t "]

= mn ■

L mnp J

(pa + pb + pc + pc + pb + pa)ti
mnp

(pa + pb + pc + pc + pb + pa)t 2
mnp

(pa + pb + pc + pc + pb + pa)t mn
^ mnp

Example:

T l2 = 2+ l+ l+ 2 synchronized with itself through 2:3:5

T,:,:, ( ,:„ = 2 X 3 X 5 (3 : 2) = 30 (3 : 2)

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PERIODICITY

2(2 + 1 + l + 2)f|

143

30

"J

= IS

(4 + 2+2+4)11

(4 + 2+2 + 4H, (4 + 2 + 2 +4)t,

> + . . . . +

30

(4 + 2 + 2 + 4)t„

*(»:»)

= io

30

3 (3:2 )t
L 30

30

]. M [ !2 ± 1 ±i± 2 t]. 1 .[a±i

30

+ 3+6)ti"

30

+

(6 + 3 + 3 + 6)t, (6 + 3 + 3 + 6)t„
H ~ r - ■ ■ - H

30

30

Tl <"> = 6 L"3o~ J = 6 L To J = 6 _

(10 +5 + 5 + 10)tj

(10 + 5 + 5 + 10)1, , (10 + 5 + 5 + 10)1, ,

+ — r ■ - - - +

30

30

30

(10 + 5 + 5+10)t«
30

In graphic representation, the resulting rhythmic series of the second order
through interference would take the following form.

Figure 20. Derivative periodicities of second order.

T S;llt {, ;J) = (4 + 2+ 2 + 1 + 1+ 2+ 3+1+2 + 2 + 4 + 3 + 1+2) +

+ (2+4+4 + 2 + 2 + 1+ 3+2+2+2 + 2+4) +

+ (3 + 1+ 2+2+2+2+3 + 1+ 2 + 2 + 1+ 3+4 + 2) +

+ (2+4+3 + 1 + 2+ 2 + 1+ 3+ 2 + 2+ 2 + 2+1+3) +

+ (4 + 2 + 2 + 2 + 2+ 3 + 1+ 2 + 2+ 4+ 4 + 2) +

+ (2 + 1+ 3+ 4 + 2 + 2 + 1+ 3 + 2 + 1+1+2+2+4).

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144 THEORY OF REGULARITY AND COORDINATION

2, Synchronization of different rhythmic series through various coefficients.

(a) The sum of the terms of one rhythmic aeries equals the sum of the terms
of another series.

Let us assume that

T' - 3 + 1+ 2 + 2+1+3-12
T* = 4 + 2+2 +4 = 12

In this case both series are ready for synchronization under any coefficient.
We shall assume now

T' : T # =3:2

The complementary factor for T' is 2 and for T ff is 3.

Thus, T'j = 3 [2(3 + 1+2 + 2 + 1+3)]
T*j = 2(3(4 + 2 + 2+ 4)]
T, =3(6 + 2+ 4 + 4 + 2+ 6)t
T"t = 2(12 +6 +6 + 12)t

T, - (6 + 2+4+4 + 2 + 6)t,+ (6 + 2+ 4+ 4 + 2+ 6)t, +
+ (6 + 2+4+4 + 2+6)1,

T*. - (12 + 6 + 6 + 12)t! + (12 + 6 + 6 + 12)t,

Figure 21. Synchronization of rhythmic series through different coefficients.

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PERIODICITY

145

(b) The sum of the terms of one rhythmic series does not equal the sum of
the terms of another series.

This case involves the double process of synchronization. In the first syn-
chronization, we equalize both series through complementary factors of the
common product. In the second synchronization, we select coefficients for the
equalized series.

Let us take T 4:3 = 3 + 1+ 2 + 2 + 1+ 3 = 12
and T 4:(2X3) = 3 + 1+ 2 + 1+1+1+1+2 + 1+ 3 = 16

T 4:3 = 12 = 3

T 4 :(2X3) 16 4

The first synchronization gives

T 4:3 = 4(3 + 1+2+2 + 1+3) = 12+4 + 8+8+4 + 12 = 48
T 4 : (2X3 ) =3(3 + 1+ 2 + 1+1+1+1+2 + 1+3) =
= 9 + 3+ 6+ 3+ 3+ 3+ 3+ 6+ 3+ 9 = 48

Now, after both series are equalized, the second synchronization through
selected coefficients can be performed. Let us designate the first series T' and
the second T", and let us use the second series twice. We shall illustrate this
synchronization through coefficients 3, 4 and 5. Then,

T' : T" : T" =3:4:5

Now we may write:

T 3 = 20(3T')
T 4 = 15(4T")
T s = 12(5T")

Substituting the values for T' and T" previously obtained, we get:

T, = 20[3(12+4+8+8+4+12)t] = 20(36+12+24+24+12+36)t =
= (36+12+24+24+12+36)ti + (36+12+24+24+12+36)t, + ...
... + (36+12+24+24+12+36)t 20

T 4 = 15[4(9+3+6+3+3+3+3+6+3+9)t] = 15(36+12+24+12 +

+12+12+12+24+12+36)t = (36+12+24 + 12 + 12 + 12 + 12+24+

+12+36)t, + (36 + 12+24+12+12+12+12+24+12+36)t 2 + . . .

... + (36 + 12+24+12 + 12 + 12 + 12+24+12+36)t, 6

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146

THEORY OF REGULARITY AND COORDINATION

T 4 = 1 2(5(9+3+6 +3 +3 +3 +3 +6 +3 +9)t] = 12(45 + 15+30+

+ 15 + 15 + 15 + 15+30 + 15+45)t = (45+15+30 + 15+15 + 15+15 +
+30+15+45)t, + (45+15+30 + 15+15+15 + 15+30+15 +
+45)t 2 + . . . + (45+15+30+15+15 + 15 + 15+30 + 15+45)1!,

The sum of each series after the second synchronization equals 48X60 = 2880.

To be represented on a 12X12 per square inch graph, using — of an inch as a

unit, this graph would require 20'.

Although such results might seem impractical at first for artistic application,
they actually are of great assistance in spatial design, particularly in covering
large areas, such as murals, for instance. In music their practicability will depend
on the speed with which they are mechanically performed. For example, if

we wanted to have the shortest unit run in — of a second (a considerable

2880 12

musical speed), the 20' of a graphed roll would take 4 minutes of performance.
This is very practical, considering that at this rate the roll would move at a speed
of 1 mile in 5 hours, 52 minutes.

F. Periodicity of Expansion and Contraction.

The periodicity of expansion and contraction determines the constant forms
of periodic variability. In general parameters it amounts to continuous increase
or decrease of spatial dimension, or of velocity in temporal dimension. In special
parameters it means continuous increase or decrease of frequency of sound waves,
which amounts to sliding pitch; continuous increase or decrease in amplitude,
which amounts to crescendo and diminuendo. In spectral hues, it means con-
tinuous transition from one part of the spectrum to another (continuous increase
and decrease in wave-length) ; continuous change of intensity of a light source; etc.

Speeding up and slowing down time values, whether in long or in short
portions of continuity, is inherent in any temporal or spatial-temporal art form.
A comparative analysis of folk art provides a large variety of illustrations. In
many a folk dance of Hungary and other countries, this variability of speed
becomes one of the most substantial thematic components.

Our perception of the visible world implies a form of spatial contraction,
namely, optical perspective. All equidistant intervals of space, in the direction
perpendicular to our field of vision, seem to contract as distance grows. After
about 1150 feet, equidistant intervals lose all distinction, and everything beyond
this distance appears "flat".

In observing "slow motion" vibratory phenomena, such as sound waves,
we find that, though all the possible frequencies within the audible range form a
mathematical and a psycho-physiological continuum, we recognize only those
that present simpler relations as harmonic ones. The series of overtones called

Digii

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PERIODICITY 147

harmonic or partial series produce a natural fraction series with regard to their
wave-length contraction and a natural integer series for the corresponding fre-
quencies. If we designate "f" as frequency and "w" as wave-length, the corres-
ponding relations will take the form of the following series:

When f = 1, 2, 3, 4, n

111 1

w = 1, -, -, -, -

2 3 4 n

The relationship of frequency to the corresponding wave-length is in square
interdependence:

f n w l:n 1

w l:n f n n 2

The decrease of wave-length in such a series produces a psycho-physiological
effect of contraction. As the difference between wave-lengths decreases math-
ematically

11111111 111 111 1

— — — — =s — ' — — — ss — • — — — = — — — — = — • etc

1 2 2' 2 3 6' 3 4 12' 4 5 20' 5 6 30'

the difference in pitch of the corresponding tones seems to approach zero, as we
hear it. 4

Thus, the natural series provide us with the full range of rhythmic possibil-
ities. Rhythms based on constant velocities are either continuous repetitions of
the terms of a monomial periodicity, or of several monomial or polynomial pe-
riodicities synchronized. Rhythms based on variable velocities are progressions
of single terms belonging to different periodicities.

Proceeding from - to -, we can produce the full scale of consecutive contrac-
2 n

tion within the limits - and -. Proceeding from - to -, we can produce the full

2 n n 2

scale of consecutive expansion within the same limits. By way of generalizing,

\_

n 111 1

we may make the following statement: In a series 2 . . - where the values

1 a b c n

a

of denominators constantly increase from a ton, the degree of expansion or contraction
increases as it approaches a and decreases as it approaches n. It depends upon the
area of the series selected for operations. For example, the degree of expansion or

contraction will be greater in the area between - < — ► - than in the area between

2 7

1 J_
7 12"

*This ia true only with frequencies not exceed- prehend as decrease of intensity instead of increase
ing about 5000 vib. per second. Further increase in pitch,
of frequencies (above 18.000 per second), we ap-

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148 THEORY OF REGULARITY AND COORDINATION

In the first case, by using all the intermediate terms of the series, we get

l + i^i+l + l + l where the differences are \ + h + h> + h + h

The initial difference in velocity fgj existing between the first two terms grows

into 77 between the last two terms, i.e., it increases 7 times (42:6 — 7). In the
42

second case we get where the differences are +

7 o 9 10 11 12 56

+ zz + + 777 + — :• The initial difference between the corresponding terms
72 90 110 132

grows from -7 to — --, i.e., it increases 27- times ( 132:56 — 2 77J.

56 132 14 \ 14/

To be applied in art, all these progressions must be expressed in the same

terms, i.e., through a common denominator. Thus, if we wish to see the expansion-
1

6 1 I 1 1 1 30

contraction 2 as a rhythmic reality, we must write — -\ 1 f- - + - = — — (-

1 2 3 4 5 6 60

1

1

+ 20151210 whcre j _ 60t t = ± and 2 = 87t.
60 60 60 60 60 1

2

Here is the graph of this rhythm :

I
t>

Figure 22. Expansion — contraction 2.

Although the natural expansion-contraction series follows the consecutive
terms of the natural fraction series, taken within selected limits, this is only a
token form of such regularity. Besides selecting the limits for 2, one can at-
tribute various other forms of regularity to the progression of its terms, using
rhythmic or other series previously described. Expansion-contractions, being

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PERIODICITY 149

combined with each other into units of a higher order, become themselves subject
to regularity, and may be treated as elements of the latter. The simplest and
most practical way of evolving a rhythmic series of deviation (various forms of
expansion and contraction) is to produce them from a constant unit added to a
unit of growth. A term (t) in a rhythmic series of deviation equals a constant
unit (t') plus a unit of growth (t) :

t = t' + T

t is subjected to regularity and varies according to the series to which it belongs.
When the series is a monomial periodic series, t is constant; otherwise it varies.

t'

The simplest and one of the most typical relations of t' to x occurs when t = —
and T i = — . The general expression for a rhythmic series of deviation is:

Tt':t = t + (f + *i) + (t' + t,) + (t' + t,) + . . . .
where t may be positive (expansion) or negative (contraction).

1. Series of deviation based on various forms of growth for t.
(a) The numerator of x grows through the natural integer series.

1

| = t+ ( t + l) + ( t , + f ,) + (, + i) + .... + (, + ^) + 2t

Example:

Figure 23. Series of deviation based on growth of t through natural integer series.

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150 THEORY OF REGULARITY AND COORDINATION

(b) The numerator of t grows through an arithmetical progression,

1

■

#. t+ (, + i) + (, + ») + (, + -) + .... + (, + ^) +

Example:

5 1
'"5

|-i + (i + i) + (i + D + 2-]+f+| +

10

5

■

i

1

r

-

T

f

-r

Figure 24. Numerator z grows through arithmetical progression.

(c) The numerator t grows through a geometrical progression.

1

¥ -, + (, + i) + (, + |) + (, + i) + ...
or, generally:

? . t + (, + =) + (, + -) + (, + ?! ) + ...

Examples :

8 ^
' ~ 8 M ~ 8

?-'+0+l)+0+i)+0+7)+0+9-

8 11 14 20 32

= -+ 1 H

8 8 8 8 8

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PERIODICITY

151

4-

ffii:::::::::=::::=d

TTT ~r T"

::::::::::::::::;:::g::::;::::;;:^::;

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

* t-

!

=E=^EEEEEE===E=====S=E===============

4

_ _i

Figure 25. Numerator t grows through geometrical progression.

(d) The numerator t grows through a summation series.

1

Tl= t'

Example:

7 1

t-, + (, + i) + (. + o + (. + o + (. + -D-

7 8 9 11 14
7 7 7 7 7

±

Figure 26. Numerator x grows through a summation series.

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152 THEORY OF REGULARITY AND COORDINATION

(e) The numerator t grows through the series of natural differences.

1

T,_ ?

? - t+ (, + ') + ( lf+ |) + (, + «) + (, + .) + ( lf + -) +
Example:

_ 9 _ 2
1 ~ 9 ~' f ~ 9

J. 1+ ( 1+ |) + ( I+ J) + ( 1+ J) + ( 1 + |) + ( 1+ ^) +

Figure 27. Numerator x grows through the series of natural differences.

(f) The numerator x grows through the series of prime numbers.

1

■ i —

t'

? , t+ ( t , + I) + (, + i) + (, + £) + (, + i) + ( t / + i) +

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153

Example:

5 1
'-5 ''"I

+(-VV(-!) + (-f)+ ■-

Figure 28. Numerator x grows through the series of prime numbers.

All the cases described here, when read backwards, represent contraction. Any
consecutive contraction may be obtained through 2 = t + ^t' — + . . .
where m may grow through any preselected series.

2. Forms of standard deviation in a binomial or a polynomial.

This form is known to musicians as rubato. We have already discussed
standard deviation as an art determinant*. Mathematically it is a special case of

t, when ' = --:. If a balanced binomial is thrown out of balance by a unit of
t'

standard deviation, we obtain:

balanced binomial B 2 = a + a

unbalanced binomial U» = (a + t) + (a + *)

or, through permutation, Uj = (a — t) + (a — t)

In other words, in order to throw a balanced binomial out of balance, it is neces-

•See Chapter 1. Section G. "Correspondence
Between Art Forms." (Ed.)

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154

THEORY OF REGULARITY AND COORDINATION

sary to add the unit of standard deviation to one of the terms of a binomial and
to subtract it from another. If T = 2t = ti + t», the unbalanced form will be

T = 2t = (tx + ^) + (t 2 + For example, if T = 10, t = y « 5,^ = ^; then

w = To + To ,n balance an To = To - + "To

1 6 4

-, or — + — out of balance.

In order to bring a polynomial out of balance, it is necessary to bring one
of its binomials out of balance. Further variations can be obtained through con-
secutive displacement of the unbalanced binomial. If we have T = mt = ti + t 2
+ ts + + t m , (where all the t's are equal) and if we select any two neigh-
boring terms as a binomial, (for instance, ti -f tj), then the unbalanced form of
the binomial will be (ti + t) + (tj — t), and the whole polynomial will be

ti + (t, + t) + (t, - t) + t 4 + . . . + t m
ti + t s + (t, + ~.) + (t 4 + t) + t 6 . . . + t m

tx + t 2 + ti + . . . + (t B _, + t) + (t m - t)

(t, + t) + (t, - t) + t, + t« + + 1„

The opposite process would bring the binomial back to balance.

Example:

2 1

t = -; t = -
8 8

_ 2ti 2t 2 2ts 2t 4

T ' = T + T + T + T

(2 l\ 2 2 3t,
Bringing the first binomial out of balance, we get T 4 =I-+-) + q + ~z + +

\8 8/ 8 8 8

, t 2 2t, 2t 4 2t, 3t 2 t 8 2t 4 2t, 2t 2 3t, t 4

+ 8 + T + T' or T' f T + 8 + T' or T + T + T + I

Or, by interior inversion of the binomial:

t x 3t 2 2t, 2t 4 2t, t 2 3t, 2t 4 2ti 2t 2 t, 3t 4

T< = 8 + T + T + T' or T + I + T + T' or T + T + 8 + T

J. O/Aer /orms o/ deviation in a binomial or a polynomial.

Any unit t may grow through any of the series previously described. This
process brings more intense expansion-contraction than through the growth of a
unit of deviation added to t. Allowing t to grow through the natural integer
series, we obtain :

2 = t, 2t, 3t, 4t, .

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PERIODICITY 155

Allowing t to grow through the summation series, we obtain:

2 = t, 2t, 3t, 5t, 8t, 13t,

t

Thus, we may produce various forms of factorial continuity previously des-
cribed. Besides this procedure, there is a possibility of varying one of the terms
of a binomial or a polynomial, while the rest of the terms remain constant or
vary in a different manner.

Here are a few of the possibilities:

Binomials

(a) The first term of a binomial remains constant, while the second grows through
any series.

Example:

1 -f 2 is a given binomial. The second terna grows through the natural series.

(1 + 2) + (1 + 3) + (1 + 4) + (1 + 5) + (1 + 6) + . . .
< >

(b) The combined form = forward-backward progression of the previous series.
Example:

(1 + 2) + (1 + 3) + (1 + 4) + . . . + (4 + 1) + (3 + 1) + (2 + 1)

(c) Both terms of a binomial grow through the same or different series.
Example:

(1 + 2) + (2 + 3) + (3 + 4 ) + or

< >

(1 +2) + (2 +4) + (3 + 8) + (4 + 16) + . . .
< >

(d) Forward-backward progression of the same series.
Example:

(1 + 2) + (2 + 3) + (3 + 4) + . . . + (4 + 3) + (3 + 2) + (2 + 1)
or

(1 + 2) + (2 + 4) + (3 + 8) + . . . + (8 + 3) + (4 + 2) + (2 + 1)

(e) Forms of interrupted expansion -con traction obtained from the previous series
through permutations.

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156 THEORY OF REGULARITY AND COORDINATION

Examples:

Theme: (1 + 2) + (3 + 4) + (5 + 6) + (7 + 8) + . . .
Var. I: (1 + 2) + (2 + 1) + (3 + 4) + (4 + 3) + . . .
Var. II: (1 + 2) + (5 + 6) + (3 + 4) + (7 + 8) + . . .
Var. Ill: (l + 2) + (7 + 8) + (3 + 4) + (5 + 6) + . . .
Var. IV: (1 + 2) 4- (2 + 1) + (3 + 4) + (5 + 6) + (6 + 5) +

+ (7 + 8) + . . .
Var. V: (1 + 2) + (3 + 4) + (2 + 1) + (4.+ 3) + . . .
Var. VI: (1 + 2) + (4 + 3) + (5 + 6) + (8 + 7) + . . .
Var. VII: (1 + 3) + (2 + 4) + (3 + 5) + (4 + 6) + (5 + 7) +
Var.VIII:(l + 3) + (3 + 2) + (2 + 4) 4- (4 + 3) + (3 + 5) +

+ (54-4) + ...

Trinomials

(a) Growth of the first term.

(l+2+3)+(2 + 2+3)+(3 +

2

4-

3)

+

(1 + 2 + 3) + (3 + 2 + 3) + (5 +

2

4-

3)

+

(b) Growth of the middle term.

(1 + 2 + 3) + (1 + 3 + 3) + (1 +

4

+

3)

+

(1 + 2 + 3) + (1 + 4 + 3) + (1 +

8

+

3)

4-

(c) Growth of the last term.

(1 + 2 + 3) + (1 + 2 + 4) + (1 +

2

+

5)

4-

(1 + 3 + 1) + (1 + 3 + 3) + (1 +

3

+

5)

+

(1

4- 3 4- 7) +

(d) Growth of the first two terms.

(1 + 2 + 3) + (2 + 3 + 3) + (3 +

4

+

3)

4-

(1 + 2 4- 3) + (2 + 4 + 3) + (4 +

8

+

3)

+

(1 + 2 + 3) + (2 + 3 + 3) + (4 +

4

+

3)

4-

(8

+ 5 4 3) 4

(e) Growth of the last two terms.

(1 + 2 + 2) + (1 + 3 + 4) + (1 +

4

4-

8)

+

(1

4- 5 + 16) +

(1 + 2 + 3) 4- (1 + 3 + 4) + (1 +

5

+

5)

+

(1

4-8 4- 6) +

(3 + 2 4- 3) 4- (3 4- 4 4- 5) + (3 4-

5

4-

6)

4-

(3

4-6 4- 7) 4-

(f) One term expands, one contracts, one remains constant.

(1 4- 2 4- 3) 4- (2 4- 1 4- 3)
(1 4- 2 + 3) 4- (2 4- 2 4- 2)
(1 4- 2 + 3) + (1 4- 3 4- 2)

Digit

Google

(2 4- 3 4- 4) 4- (1 4- 4 4- 4)
(2 4- 3 4- 4) 4- (1 4- 3 4- 5)
(2 + 3 + 4) 4- (2 + 2 4- 5)

Original from
UNIVERSITY OF MICHIGAN

PERIODICITY

157

There are many other possibilities in expansion-contraction techniques, such as
juxtaposition of various coefficients and powers. They all form rhythmic series
of the second order and are valuable in composition. 6

G. Progressive Symmetry

1. Forms of Harmonic Continuity. 7

(a) One Subject: A.

A

(b) Two Subjects: A, B.

A + (A + B) + A

(c) Three Subjects: A, B, C.

A + (A + B) + (A + B + C) + (B + C) + C

(d) Four Subjects: A, B, C, D.

A + (A+B+C) + (A + B + C + D)+(B+C + D)+D
A + (A + B) + (A + B+C + D) + (C+D)+D

(e) Five Subjects: A, B, C, D, E.

A + (A + B) + (A + B + C)+(A + B + C + D + E) + (C + D + E) +
+ (D + E) + E

•These series of the second order are treated in Each letter may be regarded as representing either
greater detail in Chapter 4. Section 6. (Ed.) a theme or a fragment of a theme. Thus, each of

'These abstract forms are more readily under- the alternatives may suggest the development of
stood if one thinks of them in relation to music. themes in a symphony. (Ed.)

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

CHAPTER 4

PERMUTATION
(Compensatory Coordination and Continuity)

A. Displacement

ANY group of elements representing different values of the same component
*~ may be expressed as a polynomial P = a+b+c+ ... + m. Such a group
can be varied by means of permutation, thus producing a modified version of
the same style.

Mechanical simplicity of variation determines the kinship of various groups.
In the closest relation to the original are groups obtained through displacement }
The latter may be regarded as a consecutive displacement of terms following one
direction. This can be performed clockwise (forward) or counterclockwise
(backward).

Original polynomial : P =(a-fb+c + ...+m)

Derivative polynomials
obtained through

displacement: Pi = b + c + . . . + m-|-a

P, = c + . . . + m+ a+ b
P, = ... + m+ a+ b+ c

P m = m+ a+ b+ c+ ...

The number of displacements equals the number of terms in a polynomial minus
one.

N d = N t - 1

A polynomial of 2 terms gives one variation through displacement; a polynomial
of 3 terms gives two variations; a polynomial of n terms gives n — 1 variations.

Displacement in a Rhythmic Group.

P<:3 = 3 + 1+ 2+ 2 + 1+ 3

Pi =1+2+2 + 1+ 3+ 3

P, =2+2 + 1+ 3+ 3 + 1

P, =2 + 1+ 3+ 3 + 1+ 2

P« =1+3+3 + 1+ 2+ 2

P, =3+3 + 1+ 2 + 2 + 1

'Displacement, a special case of permutation,
is also known as circular permutation.

158

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PERMUTATION
Graphically these displacements appear as follows:

Figure 1. Graphic representation of displacement.

Rearranged in simultaneity these displacements take the following form

Figure 2. Displacements arranged in simultaneity.

I Original from

igitized by ^.OOglC UNIVERSITY OF MICHIGAN

160

THEORY OF REGULARITY AND COORDINATION

Displacement is the basic process for evolving simultaneity and sequence in
composition. Using the same polynomial, P«.i, we may evolve the following
sequences of simultaneity through displacement.

p

ft

ft

ft

ft

ft

ft

ft

ft

p,

ft

P

P.

p.

ft

p>

p«

p*

p

ft

ft

ft

P*

P

P.

ft !

ft

ft

p<

p,

p

p,

ft

ft

ft

p

ft

ft

ft

p.

ft

p*

p

p.

ft

ft

p r

p

p.

P.

ft

ft

ft

ft

p

ft

ft

ft

ft

p

p,

Pi

ft

ft

P

ft

p

ft

ft

ft

ft

p

ft

p.

Ps

ft

ft

P.

e.t.t.

Figure 3, Sequences of simultaneity based on displacements.

Graphically this may be represented as follows:

Figure 4. Graphic representation of material in preceding Figure,

Digitized by LiOGglC UNIVERSITY OF MICHIGAN

PERMUTATION

161

The corresponding artistic expression for the sequence of simultaneity is
counterpoint. The method of displacement provides an ultimate form of rhythmic
counterpoint. Here is a typical example of rhythmic counterpoint in music:

J J J
J J J
J J J

Representing this in figures, we obtain the following:

2 + 1+1
1+2 + 1
1+1+2

Graphically this sequence may be represented as follows:

Figure 5. Rhythmic counterpoint in graph form.

f\r\cs\{> Original from

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162

THEORY OF REGULARITY AND COORDINATION

The first term displaces itself forward. Generalizing the principle of dis-
placement for the composition of sequences in simultaneity, we may state that
the number of simultaneous terms equals the number of terms in a given poly-
nomial; and the number of polynomial groups in the sequence of the first order
equals the same number (the number of terms in a given polynomial).

B. General Permutation

Permutations in general are more numerous, mechanically more complex,
and artistically a little less obvious than displacements. Nevertheless, all the
permutations of one polynomial produce modifications of one style or one national
artistic language. For example, if a certain group of numbers specifies the pitch
scale characteristic of a certain nation, all the derivative scales obtained through
permutation will be recognized as pertaining to the musical language of the
same nation. This device permits us to evolve a great many new variations on
styles already established and seemingly exhausted. The problem of creating a
new style likewise becomes an easy procedure which can be accomplished in a
much more reliable way than by mere feeling.

The number of permutations possible with a given polynomial P = a -f- b -f
+c + . . . + m equals the product of the integer numbers from one to the number
of terms of the polynomial. When P has n terms, the number of permutations
equals the product 1 by 2, by 3, by 4, ... by n.

A polynomial consisting of two terms has two permutations (including the orig-
inal).

A polynomial consisting of three terms has six permutations (including the

A polynomial consisting of 12 terms has over half a billion permutations.

There are two ways of evolving the sequences of permutations. We shall
call the first one — mechanical permutation, and the second — logical permutation.
In the case of 2 elements the mechanical permutations and the logical permu-
tations are identical:

In order to obtain mechanical permutations of 3 elements it is necessary to
take each of the permutations of 2 elements, add the third element and displace
it for each from left to right.

N p =lX2X3X..Xn.

1X2=2

original).

1X2X3=6

ab

ba

abc
acb
cab

bac
bca
cba

Digit

Original from
UNIVERSITY OF MICHIGAN

PERMUTATION

163

Logical permutations may be obtained from the first group abc by exchanging
the positions of b and c; then starting a group with b and exchanging the positions
of a and c; and, finally, starting a group with c and exchanging the positions
of a and b.

abc bac cab

acb bca cba

The displacements abc, bca and cab are symmetrically located in both of the
above groups.

abc^ bac

"-•^ abc~, bac ^-cab

acb >bca and ""-^ „-*

„-'' acb x "-bca''' cba

cab'" cba

The general method of producing a table of permutations for n elements in a
mechanical sequence requires the addition of the nth element to each group of
the table of permutations for n — 1 elements, and the consecutive displacement
of the added element from right to left. For example, if abcde is one of the group
to which the new element f is added, then f moves consecutively backwards
without affecting the sequence of the remaining terms.

abcdef
abcdfe
abcfde
abfcde
afbcde
fabcde

This holds true for every group. The mechanical sequence of permutations
produces an inverted symmetry from its center, a property already observed in
rhythmic groups resulting from the interference of uniform periodicities. 1

Examples of inverted symmetry obtained through mechanical permutation.

Example 1. Color.
a = yellow, b = blue, c = red
Y-B-R-Y-R-B-R-Y-B - B-Y-R-B-R-Y-R-B-Y

*See Chapter 3. Periodicity. (Ed.)

Digitized byGoOgle

Original from
UNIVERSITY OF MICHIGAN

164 THEORY OF REGULARITY AND COORDINATION

Example 2. Sound.

or«J b» J c« J*

J j |j>j i|j j>jJ>j J j>U j>j |jy j>|j j

/. Table of permutations in mechanical sequence.

2 Elements 1X2-2 permutations

ab ba

3 Elements 1X2X3-6 permutations

abc bac
acb bca
cab cba

4 Elements 1X2X3X4 - 24 permutations

abed

bacd

abdc

bade

ad be

bdac

dabc

dbac

acbd

bead

aedb

beda

adeb

bdea

dacb

dbca

cabd

cbad

cadb

cbda

cdab

cdba

dcab

deba

5 Elements 1X2X3X4X5 = 1 20 permutations

abede bacde
abced baced
abecd baecd
aebed beacd
eabed ebacd

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

abdce
abdec
abedc
aebdc
eabdc

adbce
adbec
adebc
aedbc
eadbc

dabce
dabec
daebc
deabc
edabc

acbde
acbed
acebd
aecbd
eacbd

acdbe
acdeb
acedb
aecdb
eacdb

adcbe
adceb
adecb
aedcb
eadcb

dacbe
daceb
daecb
deacb
edacb

cabde
cabed
caebd
ceabd
ecabd

PERMUTATION

badce
badec
baedc
beadc
ebadc

bdace
bdaec
bdeac
bedac
ebdac

dbace
dbaec
dbeac
debac
edbac

bcade
bcaed
bcead
becad
ebcad

bcdae
bcdea
bceda
becda
ebcda

bdcae
bdcea
bdeca
bedca
ebdca

dbcae
dbcea
dbeca
debca
edbca

cbade
cbaed
cbead
cebad
ecbad

UNIVERSITY OF MICHIGAN

Original from

166

THEORY OF REGULARITY AND COORDINATION

cadbe

cbdae

radeb

cbdea

caedb

cbeda

oeadb

cebda

ecadb

ecbda

cdabe

cdbae

rdaeb

cdbea

cdeab

cdeba

cedab

cedba

prdab

ecdba

dcabe

dcbae

dcaeb

dcbea

dceab

dceba

decab

decba

edcab

edcba

2. Table of permutations where some of the elements
are identical {mechanical sequence).

These sequences are evolved through the previous table. All the coinciding
groups are eliminated in their corresponding places.

3 Elements

2 identical elements. 3 permutations (= displacements).

aab
aba
baa

4 Elements

3 identical elements. 4 permutations ( = displacements).

aaab
aaba
abaa
baaa

2 identical pairs. 6 permutations.

aabb baba
abab abba
baab bbaa

2 identical elements. 12 permutations.

aabc aacb acab

abac abca acba

baac baca bcaa

caab
caba
cbaa

Digitized byGoOgle

Original from
UNIVERSITY OF MICHIGAN

PERMUTATION

4 identical elements.

5 Elements

5 permutations (= displacements),
aaaab
aaaba
aabaa
abaaa
baaaa

3 identical + 2 identical.

aabba
ababa
baaba
babaa
abbaa
bbaaa

10 permutations,
aabab
abaab
baaab

aaabb

2 identical + 2 identical + 1. 30 permutations.

aabcb

aacbb

acabb

caabb

dUdCU

dULdU

dtudU

td UdU

UddCU

UdLdU

UCddU

LUddU

aabbc

ababc

baabc

abeba

aebba

cabba

abbac

abbca

cbaba

babac

babca

bacba

bcaba

bebaa

ebbaa

bbaac

bbaca

bbcaa

il + 3.

60 permutations.

aabcd

aacbd

acabd

caabd

abacd

abcad

acbad

cabad

baaed

bacad

bcaad

cbaad

aabdc

aacdb

acadb

caadb

abadc

abeda

acbda

cabda

baadc

bacda

bcada

cbada

aadbc

aadeb

acdab

cadab

abdac

abdea

aedba

cadba

badac

badca

bedaa

cbdaa

adabc

adacb

adcab

cdaab

adbac

adbca

adeba

cdaba

bdaac

bdaca

bdeaa

cdbaa

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

168

THEORY OF REGULARITY AND COORDINATION

daabc daacb dacab dcaab

dabac dabca dacba dcaba

bdaac dbaca dbcaa dcbaa

C. Permutations of the Higher Orders.

A group (combination) of two or more elements may be regarded as an
element of the succeeding higher order. Thus, permutations can be performed
ad infinitum with any number of elements. This method is especially useful in
evolving an extended continuity from a limited number of elements.

1. Binomial elements. Binomial permutations.

Assuming that ai and bi are elements of the first order, we may evolve a
continuity through permutation.

(a, + b,) + (b, + a,)

Now let these two binomials be elements of the second order.

Then, ai + bi = a 2 and

bi + ai = b 2

Permutation of elements of the second order will produce the following continuity:
(a, + b,) + (b 2 + a,) = [(a, + b.) + (b t + a,)] + [(b, + aO +
4- (a, + b,)]

Now, ai -1- b 2 = a 8

b a + a 2 = bj

Then, (a 3 + b,) + (b, + a,) = [(a, + b 2 ) + (b, + a,)J + [(b» + a,) +
+ (a 2 + b 2 )] = {[(a, + bj) 4- (b, + a,)] + [(b, + a,) +
+ (a, + b,)]} + {[(b, + a,) + (a, + b,)] + [(a, + bi) +
+ (b, + a,)]}

Or, in general, a n = a n _, + b„_,

b n = b n _, + a n _,

Then, (a n + b n ) + (b n + a n ) = [(a„_, + b n _,) + (b„_, + a n _,)] +
+ [(b n _, + a n _ t ) + (a n _, + b„_,)]

Original from

Digitized byCjOOglG

UNIVERSITY OF MICHIGAN

PERMUTATION 169

Example:

a - 2; b = 1

(a, 4- bi) 4- (b, + aO - (2 + 1) + (1 +2)

a» >= ai -f bi = 2 + 1

b» - bi + a a « 1 + 2
(a, + b.) + (b, 4- a,) = [(2 + 1) + (1 + 2)] 4- [(1 4- 2) + (2 + 1)]

ai - a, + b« - (2 + 1) + (1 + 2)

b, = b 2 + a, = (1 + 2) + (2 + 1)

(a, + b.) + (b, 4- a,) = {[(2 4- 1) + (1 4- 2)] + [(1 + 2) 4- (2 + 1)]} +
+ {[(1 + 2) + (2 + 1)] + [(2 + I) + (l + 2)]}.

A two-element combination may be chosen from any group. A trinomial
in its six variations may be assumed an element of the second order. Thus,

ai + bi + ci = a s bi + ai + ci = di
ai -I- ci 4- bi = b s bi + ci + ax = e 2
ci + ax 4- bi = cj cx + bx + ax = f 2

The following 30 binomial combinations are possible from these 6 elements:

a t + b s

b 2

4- a 2

c 2 4- a 2

a 2 + cj

b,

4- c 2

c 2 4- b 2

a s + dj

b 2

4- d 2

c 2 4- d 2

a 2 + d

b,

4- e 2

c 2 4- e 2

a 2 + fj

b 2

+ f 2

c 2 4- f 2

dj 4- aj

ei

4- a 2

f 2 4- a 2

d, 4- b,

e 2

4-b 2

f» 4- b 2

d 2 4- cj

e 2 4- cj

f 2 4- c 2

d, 4- e 2

e 2

4- d 2

f 2 4- d 2

d, + f,

e 2

4-f,

f 2 4- e 2

By continuous inversion of these binomials in their consecutive orders, we
obtain the binomial growth of trinomials.

rv -■ _j l r\rscs\f> Original from

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170 THEORY OF REGULARITY AND COORDINATION

Let us develop a 2 + fi where

aj = ai + bi + ci and
f 2 = Ci + bi + ai

Thus, ai = a 2 + f 2 and

bs — fi + a 2

Substituting at and fj for their trinomial expressions, we obtain:
a, + b, = (a, + f 2 ) + (f, + a,) = (ax + bi + d) +
+ (ci + b, + ai) + (ci + b x + ai) + (ai + bi + Ci).

Likewise, a 4 = a» + b$ and

b 4 = bj + a»

Thus,

a 4 + b 4 = (a, + b,) + (b, + a,) = [(a, + f.) + (fi + a,)] +

+ [(fi + a,) + (a, + f,)] = {[(a x + b, + d) + (c, + bx + a,)] +
+ [(c, + bi + ai) + (ai + bi + Ci)]} + {[(ci + bi + a,) +
+ (ax + bx + c,)] + [(a, + bx + ci) + (ci + b, + ax)]}.

Assuming ax = 3, bx = 2, ci = 1,

we obtain a» = 3 + 2 + 1

U = 1 + 2 + 3

a, = a 2 + f 2 = (3 + 2 + 1) + (1 + 2 + 3)
b 3 = f, + a 2 = (1 + 2 + 3) + (3 + 2 + 1)
a, + b, = [(3 + 2 + 1) + (1 + 2 + 3)] + [(1 + 2 4- 3) + (3 + 2 + 1)]
a 4 = aj + b s
b 4 = b» + a»

a« + b 4 = {[(3 + 2 + 1) + (1 + 2 + 3)] + [(1 + 2 + 3) + (3 + 2 + 1)]} +
+ {[(1 + 2 + 3) + (3 + 2 + 1)] + [(3 + 2 + 1) + (1 + 2 + 3)]}.

In general, any polynomial (a + b + c + . . . + m) may become an element
of the succeeding order and may be combined with other polynomials. It is
not necessary that the polynomials P' and P", etc., be derivatives of P. They
may belong either to the same or to a different series. Thus, any trinomial and
its variations produce 6 elements of the following order. These 6 elements may
combine by 2, by 3, by 4, by 5 and by 6.

rv -■ nnnlp Original from

Digitized by ^WJJ^ UNIVERSITY OF MICHIGAN

PERMUTATION

171

Three elements of the first order combined by two, produce, through bi-
nomial permutations, 12 terms.

ai + bi + ci = a 2
ai + ci + bi = b s

a 2 has three terms.

a » + bj + bj + aj yields 12 terms.

Three elements of the first order combined by three, produce, through tri-
nomial permutations, 54 terms.

ai + bi + Ci = aj
ai + ci + bi = bt
ci + ai + bi = ci

a 2 has three terms

btf tf M

2

M tt tt

Ci

a2 + bt + C2 yields 9 terms.

6 (3 X 3) =54

The increase in the number of combined elements causes rapid increase in
the number of permutations. Thus, by combining trinomials of the first order
by 6, we obtain six elements of the second order. They produce 1X2X3X4
X 5 X 6 = 720 permutations. Each element is a trinomial in this case. This
yields in the second order

3 X 720 = 2160 terms.

In general, any polynomial of the first order, ai + bi + Ci + .... + mi
produces through permutations a2, b2, C2 ... m t . These being combined by 2,
by 3, by 4,. . . by n, yield respectively, 4, 6, 24, ... or 1 X 2 X 3 X ... X n
term groups. The latter become elements of the third order. Expansion of the
number of elements in their consecutive higher orders grows, through per-
mutation, very rapidly into an enormous number of terms. For example, 3
elements of the first order produce 6 elements of the second order, 720 elements
of the third order, etc.

The artistic value of this method lies in the production of simpler groups
through their higher orders. Such continuity achieves the effect of variety in
thematic unity.

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

172 THEORY OF REGULARITY AND COORDINATION

D. Mechanical Scheme for the Permutation of Four Elements

/

r,

-* ,' n

[£lt u£i LUitui \'

, /
X

/ » v

4 ■ >

uuuumiiiii

F

1 Tl ,' '

» m

v «'■ »r

JIJ LLUILU Ilij

B i

UULLUHUIiU

liLiuuiiumj

UJJ UU I1LI tlU*

8.

B

o ' i

Bt

1 71 1 «

ifuuu UiJ uii

tiif un uii iiij

UUtLU UU

tLLI llLl UiJ LL2J
C

liuuiimiiiij \/

^uijiiijiiuiin

ujJuiitiuiiii

iiu Lujtiu ui

Figure 6. Mechanical scheme for permuting four elements.

rv - ■ c\r\cs\{> Original from

Digitized by K^wglK. UNIVERSITY OF MICHIGAN

"From shelf to crossbar to cage-top
and back again the gibbon swarmed,
breaking down rhythm and timing
into their cube roots and building
them up again, without so much as a
hairsbreadth of mis judgment."

"With the Greatest of Ease,"
by Paul Annixter.

CHAPTER 5

DISTRIBUTIVE INVOLUTION
(Coordination and Continuity of Harmonic Contrasts)

A. Powers

ALGEBRAIC powers are thebasisof factorial-fractional continuity. They build
■*■ *■ rhythm within a unit (a bar in music, an area in design) and organize the
whole continuity rhythmically.

The power formulae found in standard algebra do not express the distributive
properties of a resulting polynomial, which are.most important in the construction
of an art form. 1 They may be used although they are not quite satisfactory be-
cause of

(1) the limited number of terms in the resulting polynomials;

(2) the extremely unbalanced values of the terms; and

(3) the lack of distinct representation of qualities, values and relation-
ships in the resulting terms and groups.

The following formulae establish full correlation between the terms and the
groups of a polynomial according to the initial ratio. Raising to power should be
performed through consecutive multiplication of all the terms of a preceding
power by each term of the given polynomial. Thus, the factoring coefficients are
related to each other as the terms of the initial polynomial.

Example :

(a + b) 2 = (a* + ab) + (ab + b 2 ).

Here a 1 is related to ab, and ab is related to b 2 as a is related to b. a 1 + ab is
related to ab + b l as a to b.

!The distributive properties are treated in com-
binatory analysis.

Digitized by GoOgle 173

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UNIVERSITY OF MICHIGAN

174 THEORY OF REGULARITY AND COORDINATION

a
b

a =

a*
ab

ab
b 2

2; b

4

2

2
1

a 1 + ab
ab + b*

= 1

4 + 2
2 + 1

(1) The First Power
Factorial monomial periodicity with one determinant, x.

F x = xti -f xt 2 + xt s + + xt x

Factorial binomial periodicity with two determinants, x and y.

F x+y = (xt, + yt 2 ) + (xt s + yU) + (xt 6 + yt«) + . . . + (xt x+y _, + xt x+y ).

Factorial trinomial periodicity with three determinants, x, y and z.
F x+y+I = (xti + yt 2 + zta) + (xt 4 + yt» + zt 6 ) + (xt 7 + yt 8 + zt„) + . . .
. . . + (xt x+y+i-2 + yt x+y+z-l + zt x+y+i)-

Factorial polynomial periodicity with n determinants, a, b, c, .... m,
where a + b + c + ... + m = k.

F a+b+c+ . . . +m = (ati + btj + ct, + . . + mt„) + (at n+ , + bt n+2 + ct n+3 + . . .
. . . + rntij + (at 2n+ , + bt 2n+2 + ct 2n+3 + . . . + mt 3n ) + (at 3n+ i +
+ bt 3n+2 +ct 3n+3 + . . . + mt 4n ) + . . . + (at k _ n + bt k _ n+ , 4- ct k _ n+2 + . . .
. . . + mt k ).

Fractional monomial periodicity with one determinant, —

x

,,_!■+*■ + »• + ...+«!

Fractional binomial periodicity with two determinants,

x 4- y

and

x4^ + x+^ \x+y x+y/ \x+y x+y/ \ x+y x+y/

x+y x+y

Digit

x+y x+y/ \ x+y

f\r%oit> Original from

UU £V UNIVERSITY OF MICHIGAN

x+y

•x+y

DISTRIBUTIVE INVOLUTION

175

Fractional trinomial periodicity with three determinants,

and

x+y+z x+y+z x + y + z

+ _^=(^V + ^V+^V)+(-^V+

T^+i + 7+7Tz + T^Tz \x+y+z x + y+z x + y+z/ \x + y+z

_|_ yti _|_ zt « \ _|_ + / xt x+y+1 _ 2 yt, +y+ ,_! zt x+y+x \
x + y + z x+y+z/ \x+y + z x+y + z x+y+z/'

Fractional polynomial periodicity with m determinants,

a b c m

a+b+c+...+m a+b+c + ...+m a+b+c + . . . +m ■ • • a+b+c + . . . +m

B+b+ c+... +m ^ / ati bti ^ ct> ^

a \a+b+c+...+m a+b+c + ...+m a+b+c+...+m

a+b+c+. . .+m

, f™^ at "+' bt„+ 2

a+b+c+. . . +m/ \a+b+c + . . . +m a+b+c + . . . +m

a+b+c+...+m a+b+c + ...+m/ \a+b+c + ...+m

)+('

+ . . .+m/ \c

ct n+1 , , mt 2n ^ , ( at 2n +l

bt 2n +2 _j Ct 2n +3 , mtj n

a+b+c+...+m a+b+c+...+m a+b+c + ...+m

| btk-n+l | ct lc-n+2 j_

+m a+b+c+...+m a+b+c + ...+m

/ at k _ n
\a+b+c+. .

)

a+b+c + . . . +m/

Here n is the number of determinants a, b, c, ... m and a + b + c + ...
. . . + m = k.

In the following formulae, the expressions with n determinants will be sim-
plified as follows:

k

F = (at, + bt 2 + . . . + m n ) + + (at k _ n + bt k _ n+ , + . . . + mt k ), and

k at, bt2 mt n atb_ n btt-n+i mt k

I

k k k
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UNIVERSITY OF MICHIGAN

176 THEORY OF REGULARITY AND COORDINATION

(2) The Second Power

F x 2 = x J ti + x 2 t, + x*t, 4- + X l t x 2.

F (x+y)2 = (x J t, + xyt,) + (xyt, + y J t 4 ).

F (x+y+l) 2 = (x*ti + xyt, + xzt,) + (xyt« + y't s + yzt,) + (xzt 7 + yzt, + z*t,)

F (a+b+c+ . +m) 2 = (a'ti + abtj + act, + . . . + amt n ) + (abt n+1 + b't n+2 +

+bct n+3 + + bmt 2n ) + . . +(amt n 2_ n + bmt n 2_ n+ ,+cmt n 2_ n+2 + .

. . . + m s t„2).

General Formula of the Second Power of a Polynomial, Preserving the Initial

Distribution of its Terms.

(a + b + c + . . . +m) 1 =a(a+b + c + ...+m)+b(a+b+c + ...

. . . + m) + c(a +b + c + . . . + m) +. . . + m(a + b + c + . . .+ m)

- (a 1 + ab + ac + . . . + am) + (ab + b l + be + . . . + bm) +

-f (ac + be + c 1 + . . . + cm) + . , . + (am + bm + cm + . . . + m 1 ).

(i) 1 X* X 1 X 1 X*

f - r xHi i xytt 1 1 r xyt> i ytu 1

(ife+ife) 1 L(x + y)» ^ (x + y)»J T L(x + y)» T (x + y)«J

f r ^ + ^ + ^ — 1+

'(i^+^+i^.) 1 = L(x + y + z)» ^ (x + y + «)» (x + y + «)«J
"■"Lfx + y + z) 1 ^ (x + y + zY ^ (x+y + z)'J T Lf "

(x+y + z)» (x + y + z) 1 (x+y+z)M L(x + y+z)*

. yzt, z't, I

"•" (x+y+z) 1 " 1 " (x+y+z) 1 ]

f,» b c rn, 2 /a l t, abt, act, . amt,\ / abt n+1 b't„+ 2

, bct n+3 . bmt 2n \ / act 2n+ i bct 2n+2 c't 2n+3

+ — + • • • + + + ~k^~ + icy + • •

, cmt 3l A / a mt n 2_ n b mt n 2_ n4 . 1 cmt n 2_ n+2 m 2 t„2 \

■■■ + + V"~kT~ + "~kT— + — ^r— + • ■■+ v )

rv -■ __j l nnnlp Original from

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DISTRIBUTIVE INVOLUTION 177
(3) The Third Power
- x»ti + x*t, 4- x»t, 4- .... 4- x'tj

F ( , +y) 3 = [(xHi +x«yti) 4- (x'yts + xy l t 4 )] + [(x'yt, + xy»t.) + (xy"t T + y»t,)]

F(« +y+l) 3 = [(x»ti + x*yti + x'zts) 4- (x*yt« + xy*t 6 + xyzte) 4- (x*zt 7 +

+ xyzt 8 + xz't.)] + [(x*yt 10 4- xy»t„ + xyzti,) + (xy«ti, + y*t u +

+ y*2tii) + (xyztu + y'ztn 4- yz'tu)] + [(x J zti, + xyzt ao + xz*t,i) +
+ (xyztti 4- y'ztj, + yz't,*) + (xz*t„ + yz*t ie + z*t 17 )]

F ( . +b+c+ ... +m) 3 = [(a»t, 4- a J bt 2 + a'ct, + ... + a*mt n ) + (a l bt n+ , +
4- a , b 1 t n+2 + abct n+3 + . • . + abmt 2n ) + (a I ct 2n+ , + abct 2n+2 +
+ ac*t 2n + 3 + . • . +actm 3n ) + . . . 4-(a , mt 3n +i + abmt 3n+2 +acmt 3n+3 4- • . .
... + a'm'tta)] + [(a , bt 4n+ i 4- ab»t 4n+2 4- abct^+3 4- ... 4- abmt Jn ) +
4- (ab*t 5n+ , 4-'b»t 5 „ +2 4- b*ct 3 „+ 3 4- ... 4- b'mO 4- (abct« n+1 +
4- b'ct^n+2 4- b'c'tan+j + bcmt 7 „) + . . . + (abmt 7n+ , 4- b J mt 7n+2 4-
4- bcmt 7n+3 4- ... 4- bm'tg,,)] 4- [(a J ct 8n+ , 4- abct 8n+2 4- ac l t 8n+3 + - . .
. . . + acmt^n) 4- (abct^+j + b'ct^+j 4- bc't^+j 4- ... 4- bcmt 10 „) +
4- (ac J t,on+i 4- bc , t, 0n+2 4- c*t m+3 4- ... 4- c'mtnJ 4- . . . 4-(acmt„ n+ , 4-
4- bcmt Un+2 + c'mtim+j + . . . 4- cm l t nn )] 4- ... 4- [(a l mt n 2_ 4n +
+ abmt n 2_ 4n+ , 4- acmt n 2_ 4n+2 4- . - . 4- am't,^,,.,) 4- (abmt n 2_ 3n 4-
4- b l mt n 2_ 3n+l 4- bcmt n 2_ 3n+2 4- . . . 4- bm l t n 2_ 2n _,) 4- (acmt n 2_ 2n 4-
4- bcmt n 2_ 2n+ , + c'mt n 2_ 2n+2 4-. . . 4- cm 2 t n 2_„_,) 4-. . . 4- (am l t n 2_ n 4-
4- bm 2 t n 2_ n+1 4- cm'tnJ.n+j 4- 4- m»t n 2)]

General Formula of the Third Power of a Polynomial, Preserving the Initial

Distribution of its Terms.

(a + b + c + . . . + m)« - [(a» +a'b + a'c + ... I a'm) 4- (a*b +
4- a 2 b l 4- abc 4- ••• 4- abm) 4- (a*c 4- abc 4- ac J 4- • ■ • 4- acm) 4- • ■ ■
... 4- (a*m 4- abm + acm + . . . 4- a J m J )] 4- [(a l b 4- ab» 4- abc 4- . . .
... 4- abm) 4- (ab l 4- b' 4- b J c 4- . . . 4- b»m) 4- (abc 4- b J c 4- bV + . . .
... 4- bcm) 4- ... 4- (abm 4- b'm 4- bcm + • • . + bm 1 )] 4- [(a*c 4- abc 4-

r\r\Ci\{> Original from

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178 THEORY OF REGULARITY AND COORDINATION

+ ac* + . . . + acm) + (abc + b l c +bc 2 + . . . + bcm) + (ac 2 + be 1 +
+ c' + .... + c J m) + . . . + (acm + bcm + c 2 m + . . . + cm 2 )] + . . .
. . . 4- [(a 2 m + abm + acm + . . . + am 2 ) + (abm + b 2 m + bcm + . . .
. . . + bm 2 ) + (acm + bcm + c 2 m + • • • + cm 2 ) + • ■ • + (am 2 +
+ bm 2 + cm 2 + . . . + m»)]

f - U + 1 2 + h . . k

f = [Y x>ti + J^J] + ( , xy't* VI .

(^ y +ife) 3 LV(x + y)» (x 4- y)'/ ^ \(x 4- y)» ^ (x + y) V J
I"/ x 2 yt» xy 2 t, \ / xy 2 t 7 y't, \1

+ L\(x + y)» (x + y)v \(x + y) 1 (x + y) V J

{ _ 17 x>t * - x *y t2 , x ' zt » ^ ,

L\(x + y 4- z)» (x 4- y + zY ^ (x + y 4- z)»/

/ x 2 yt< xy 2 t t xyzt 6 \ / x 2 zt 7

+ \(x 4- y + z)» (x + y 4- z)» (x + y 4- z)V \(x + y + z).'

i xyztg , xz 2 t» \"| I"/ x 2 ytio xy 2 t n
+ (x + y 4- z)« (x 4- y 4- z) V J LV(x + y 4- z)« (x + y 4- z)«

. xyztn \ / xy 2 tn y'ti 4 ' y 2 ztu \

+ (x + y + z)V \(x+y+z) >t (x + y+z)' + (x + y+ z)V +

. ( xyzt " , y* zt " yz 2 tis \~| |Y x 2 zt u

+ \(x + y + z)» ~*~ (x + y 4- z)» + (x + y 4- z)»/J L\(x + y 4- z)» +

xyztao xz 2 t 2 i \ /__xyztjj_ y 2 zt i8

(x+y + z) 1 (x + y 4- z)V \(x + y + z) 1 i " (x + y 4- z) 1 +

+ (x4y+ z)V + \(x+y+z) I+ (x + y+z)' + (x + y+ z)«/J

Ka'ti a 2 bt, a 2 ct, a 2 mt n \ / Vbt n+1

a 2 b 2 t n+2 , abct n+3 abmt^N / a 2 ct 2n+l abct 2n+2

~ "l^ - + ~" k* - + • ■ ■ + k« r vTr- + ~" \S~ +

:mt 3a \ i i / a'mt 3H i ^ abmt 3n+2 ^ acmt 3n+3
k* / V k* k* k*

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UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION 179

/ ab*t 5n .n b*t Sn+2 b'ct 5n + 3 h'tnt^A / abct^

V k 1 k» k» k» / V k'

b'ctftn+j b'c^n+j + bcmt 7t A / abmt 7n +i +

k 1 k* k* / V k*

: » ArlA k» + v

'Wn^ + ^ abctgn

■ bcmt 10n \ / acHigBHM bc't 10B +2 c'tmn+j | ^ c'mtn n \ }

k» / \ k* k* k 1 k* /

, ac^gn+a acmtonN / abct, n+1 b'ct 9n+2 , bc't^+a

+ — tt— + ... +— — J + l .. + k> + k> +

/ acmt lln+ i bcmtitn+2 c*mt lln +3 + + cmHigA I +
\ k* k* k* k* / J

... + l

Ka t mt n 2_ 4n abmtnZ-t.H.! acmt n 2_ 4n+2 , , am t t n 2_ 3n _, > \
~kr- + ^ + si +••+ k . ; +

/ abmt n 2_ 3n b t mt n 2_ 3n+1 bcmt n 2_ 3n4 . 2 + + bm't n 2_ 2n _A +
V k* k» k* k* /

/ acmt n 2_ 2n bcmt n 2_ 2n+1 c t mt„2- 3 „. t .2 bm 2 t n 2- 2n -A
+ \ k» k« k» ' k» /

/amVn . bm t t n 2_ n+ i cm't n 2_ n+2 m l t n 2\~|

+ + ~ + — + ■■ + -^~)}

(4) Generalization. The nth power.
F x n = x n t! + x n t, + x n t, + . . . + x n t x + . . . + xV.

B. General Treatment of Powers

Besides the technique of raising to powers through consecutive multiplica-
tion of all the terms of a polynomial by each term, it is possible to raise any poly-
nomial to any power directly. This problem consists of three items:

1. To find the general number of terms in a given power.

2. To find the progression in which they are distributed.

3. To find the values of the terms.

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180 THEORY OF REGULARITY AND COORDINATION

1. Binomials.

The number of terms of the nth power of a binomial equals

N t = 2 n

The distribution of terms of the nth power in any given order equals

D, - 1 + 2 + 1

The coefficients determining the number of groups in their orders respectively
are

V c - 1, 2, 2», 2». .

,n-2

These coefficients increase as the orders of groups decrease. The group of the
first order has the coefficient 2 n_2 ; the group of the nth order (the whole) has
the coefficient 1.

Now we can write the complete expression for the distribution of the terms
of the nth power of a binomial.

iya fti on
= -+-+-

2» 2 2*

/ 2 n 2 " 2"\

~ 2 {j> + T> + T>)

(T 2 n , 2 n \

+ » +

(2 n 2" 2 n \
? + ^ + = 2 n " 2 (1+2 + 1)

2" =

Let x and y be the terms of a binomial and a, b, c . . . the terms of the nth
power. The major term, in the formula for distribution, doubles a single term or a
group of the power polynomial at any given order.

Thus if n = 2

I 1

(x + y) 1 = a + b + b + a

Here a and b are distributed in the progression 1+2 + 1, where b taken

2 , 21 22

twice represents the major term ; 2 n =1 and — +— +— =1+2 + 1

m m* Mt

Digiti

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UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION 181

Likewise if n = 3

i 1 i 1

(x+y^a+b+b+c+b+c+c+d

t I

2» 2* 2*

2* = — +— + — = 2 + 4 + 2, where the major term is a binomial b + c

taken twice.

2" = 2 + ?| + = (1 + 2 + 1) + (1 + 2 + 1), representing the groups

of the first order.
If n =4, then

I "" 1

i 1 i 1 i 1 i 1

(x + y) 4 = a+ b+ b+ c + b+ c+ c + d+ b+ c+ c + d+ c+ d + d + e

i i

2< 2* 2*
2'- r , + 7 + ? =4+8+4

(2* 2* 2*\
2~, + J, + J t ) - (2 + 4 + 2) + (2 + 4 + 2)

2< = 21 (l< + |! + ^) = (1 + 2 + !) + (1 + 2 + !) + <* + 2 + ^

If n = 5, then

I 1

i 1 i 1 i 1 i 1

(x+y)*=a+b+b+c+b + c+ c+ d+ b+ c+ c+ d+ c+ d+ d +

I I I I

I

I 1

i 1 i 1 i 1 i 1

+e+b+c+c+d+c+d+d+e+c+d+d+e+d+e+e+f
■ ■ i i

2» 2* 2*

= 21 + 7 + ^ = 8 + 16 + 8

/2 s 2* 2 6 \
= 2 {- + - + -J = (4 + 8 + 4) + (4 + 8 + 4)

2 l = 32

= 2i (f* + Y* + f) = (2 + 4 + 2) +( 2 + 4 + 2 ) +(2+4+2) +(2+4+2)

= 2t (j>+f<+f) = (1+2 + 1) + d+2 + l) + (1+2 + 1) +

+ (1+2 + 1) + (1+2 + 1) +(1+2 + 1) + (1+2 + 1) + (1+2 + 1)

rv - tvt k c\r\cs\c> Original from

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182 THEORY OF REGULARITY AND COORDINATION

When the distribution of the terms at the nth power has been found, their
value can be obtained directly from the following formula.

(x + y) n = x n + x n_1 y + x n - 2 y 2 + x n_3 y 3 + . . .
. . . + x 3 y n_3 + x 2 y n " 2 + xy"- 1 + y n .
Here x", x n-1 y. etc. correspond to the terms of distribution a, b, c, . . .
a = x n ; b = x n_1 y; c = x n-2 y 2 ;

Examples:

(x +y) s = a + b + b + c

a = x 1 ; b = xy; c = y*
(x -f y) 1 = x 1 + xy + xy + y 1

(x + y)«= a +b+b+c+b+c+d
a = x 1 ; b = x J y; c = xy 2 ; d = y 1
(x + y)' = <* + x*y + x*y + xy 1 + x'y + x'y + xy 1 + xy 1 -f y'

(x+y) 4 =a+b + b+ c+ b+ c+ c+ d+ b+ c+ c+ d+ c +

+ d +d +e
a = x 4 ; b = x'y ; c = x J y s ; d = xy* ; e = y 4
(x + yj 4 = x 4 + x'y + x'y + x'y 2 + x'y + x l y 2 + x*y 2 + xy' +

+ x'y + x*y* + x J y s + xy' + x*y 2 + xy' + xy' + y 4

x = 3 ; y = 2 ; n = 5
(3+2)*=a+b+b+c+b+c+c+d+b+c+c+d + c + d +
+ d+ e+ b-fc+c+d+c+d+d+e + c + d + d +
+e+d+e+e+f

a = 243; b = 162; c = 108; d = 72; e = 48; f = 32
(3 + 2)' = 243 + 162 + 162 + 108 + 162 + 108 + 108 + 72 +162 + 108 +
+ 108 + 72 + 108 + 72 + 72 + 48 + 162 + 108 + 108 + 72 +

+ 108 + 72 + 72 + 48 + 108 + 72 + 72 + 48 + 72 + 48 + 48 + 32.

rv sti f\r\n\{> Original from

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DISTRIBUTIVE INVOLUTION

183

The fractional form of a power polynomial can be obtained from the same
formula by using values for the numerators and the sum of the binomial at its
nth power as the denominator.

/ x y V x" x"-'y x"-y x^V

\x + y + x + y/ " (x + y) n + (x + y) n + (x + y) n + (x + y) n + ' ' '

x 3 y n - 3 xV" 2 xy"" 1 y n
. . . H 1 1 1-

/..i ..\n 1 /.. I _.\n 1 /.. i _.\n 1

(x + y) n (x + y) n (x + y) n (x + y) n
Example:

/ 3 2 V _ /3 2\> _ UZ_ UO^ 162_ _108_ 162_

\3 + 2 + 3 + 2/ \5 + 5/ ~ 3125 3125 + 3125 + 3125 + 3125 +

108 108 , _72_ U>2_ \08_ ]08 72_ J08_ _72_
3125 3125 3125 3125 3125 3125 3125 3125 + 3125

4_ + _i*L i^l J08 _72_ m 8 . Jl_

3125 3125 + 3125 3125 3125 3125 + 3125 3125 + 3125 +

, 48 , 108 , 72 , 72 , 48 72 , 48 , 48 , 32

+ ~ttzz + ttt; + rrr; + ttt; + — — + t— + — — + ttt7 +

3125 3125 3125 3125 3125 3125 3125 3125 3125

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

The discovery of new laws of un-
stable equilibrium is the end that
will be attained by the innovator
who reveals new paths of art."

From The Biological Bases of the Evolu-
tion of Music, by Ivan Kryzhanovsky.

CHAPTER 6

BALANCE, UNSTABLE EQUILIBRIUM AND
CRYSTALLIZATION OF EVENT

STRUCTURE is in a state of stable (stationary, static) equilibrium when

The density of such a structure becomes its potential, bearing the tendency
of unstabilization. The density set of a ratio can be expressed as a fractional
form of unity:

The greater the value of n, the denser the set. And the denser the set, the greater
the stability of its potential.

To illustrate, let us take R = § . In this case stable equilibrium of the two major
components can be expressed as £ + =SE. The potential of unstabilization
in this case equals i.e., the fractional determinant of the f series. As each of
the components in this particular case has the value of its potential (i.e., the
value of the component equals that of the potential), or j = \, the potential may
be regarded as one that is insufficient to change the state of stable equilibrium.

To increase the power of the potential, we must increase the density of the
set. This can be accomplished by means of involution. By squaring the general
determinant of a series, we may produce a denser set and a more powerful poten-
tial (the fractional determinant).

In the case of \ series, where \ is the general determinant and \ its fractional de-
terminant, (the latter being also its potential of unstabilization), the squaring of
its general determinant yields:

the ratio of its two major components equals one:

R = 1

Digit

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UNIVERSITY OF MICHIGAN

BALANCE, UNSTABLE EQUILIBRIUM

185

The potential of unstabilization becomes \, and unstable equilibrium acquires
the following form:

ttp» 2 + 12-1 31

UE'j. = — 1 — = - + - and

4 4 4 4

UE 10 - 4 + 4 ~ 4 + 4

E'i« is achieved by adding the potential to the first term and subtracting it from
the second (i.e., by shifting the potential in the negative direction of its own
value), and E"i» — by reversal of the first operation. The resulting unstable
equilibrium fluctuates between E' and E":

R,(EV — E",) -(] + l) tl +(l + |)t f .

Still further refinement of the form of unstable equilibrium can be accom-
plished by greater condensation of the original set$. Third power involution, i.e.,
cubing of the general determinant, provides such a means. In our case, unstable
equilibrium of the second order assumes the following form :

/2\» 8 4 4

SE =1-) = " B - + -• As the potential of stabilization is the fractional
\2/ 8 8 8

determinant, it equals -. Then:

R.2» = (E'2

E '">=(hiR+i)

This procedure can be extended ad infinitum. A set can be made as dense as
desired, and the potential of stabilization — as powerful as desired. The set
(general determinant of the series) becomes saturated quickly and reaches a
limit beyond which instability becomes imperceptible. The general form of the
resulting unstable equilibrium appears as follows:

„„ t /n + 1 n - l\ /n - 1 n + l\

At the average velocity of the last 100,000 years, the chronological life-span
of the power-index must be counted in the tens of thousands of years. For ex-
ample, the £ series, as it manifests itself in the form of unstable equilibrium of

Digitized by GOOgle

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UNIVERSITY OF MICHIGAN

186 THEORY OF REGULARITY AND COORDINATION

the duration-groups in music, is at present at least 30,000 years old. Unstable
equilibrium of the f- series, as it manifests itself in the field of graphic symbols,
as well as duration-groups in music, can be traced several millennia back: to the
"mogen dovid" of the Hebrews, and the triadic measure, "Divine Trinity"
symbol of early Catholic liturgical music.

We can look upon the involution, and in some cases factoring, technique as
a hidden mechanism behind the morphological evolution of the forms of unstable
equilibrium. As in the field of organic phenomena, this evolution is slow in pace.
Once an adaptable form has been developed, it acquires a better chance for long
survival, like the "fittest" of bio-zoological forms.

An event may be considered crystallized when it reaches its optimum during
the period of observation. Crystallization means that the organizational tendency
has acquired its maximum of realization. Crystallized events are rational; they
are a part of the continuum of eventuality and produce a series of isolated terms
between which are all the events undergoing the process of crystallization. The
latter are irrational and may be considered as "eventual" states of the process
of crystallization. The irrational polygon forms that gravitate between the form
of a triangle and the form of a square may be looked upon as a "would be"
triangle (eventual triangle), or a "would be" square (eventual square), depending
on the form of tendency of the eventuality (in this case, variation of the angle
value).

An event spends itself during the period of crystallization. An identical event
can be duplicated only if all specifications are scientifically integrated. The record
of an event, however, can duplicate the event to the degree of precision that
physical conditions permit. Up to the present, scientific specifications cannot be
worked out f:>r the "fatal" event. These can be reproduced only from the record,
which is either physical (optical, acoustical) or psychological (mnemonic).

A. Formulae

1

unity

fractional equivalent
of unity

t t

stable equilibrium

u

unstabilizer

t

UE =

jt + u £t

u

unstable equilibrium

t

R

it + u
it-u

* 1

ratio of UE

Original from
UNIVERSITY OF MICHIGAN

BALANCE, UNSTABLE EQUILIBRIUM . . .
Summation:

t - 2\i t + 2u t 3t + 2u 5t + 2u 8t + 4u 13t + 6u
2t 2t t' 2t 2t 2t 2t

21t -|- lOu 34t + 16u 55t + 26u 89t + 42u 144t + 68u
2t 2t 2t 2t 2t

SE — stable equilibrium
UE — unstable equilibrium

2 OT , 1 1
- SE - - + -
2 2 2

No UE

/2V 4 2 2 _

\2/ 8' 8 8

5 3

i + i

2V 8 4 4

SE = - + -

_ _ 5 3
UE = - + -

(-Y - -

\2/ 16'

SE
UE

8

+

8

16

16

9

+

7

16

16

\2/ 32 32 32

32 32

fV = 64 . S£ = 32 , 32
<2/ 64' 64 64

, Tr > 33 31

UE 1

64 64

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

THEORY OF REGULARITY AND COORDINATION
| NoSE

3 3

(s)'4 NoSE

""MS

/3\ 4 81

(3) -M N ° SE

41 . 40

5'

No SE

/5V _ 25
\5/ "25 :

/5\» = 125
\5/ ~ 125

UE =

No SE
UE =

No SE
UE =

2+2
5 5

13 12
25 25

63_ 62_
125 125

-• NoSE
7

/7V = 49
\7/ ~~ 49 1

No SE

UE~^
49 49

in r^nnnlf* Original from

gilizeti by VjUU^I^ UNIVERSITY OF MICHIGAN

BALANCE, UNSTABLE EQUILIBRIUM

189

10

io :

SE =

— + —

10 10

UE =

10 10

11
11 :

No SE

UE =

11 11

12

12 :

SE =

— + —

12 12

UE =

12 12

B. Unstable Equilibrium in Factorial Composition of Duration-Groups

First Form: in composing the resultants of interference of two synchronized
biners, select two identical major generators (a) and two non-identical minor
generators (b and b') which are the successive terms of one summation-series,
and the sum of which equals the value of major generator (a) ; the minor genera-
tor (b) of the first biner is greater than the minor generator (b') of the second
biner, or vice-versa.

Formula:

Examples :

E' = r a+b + r a+ b' and /° r

E" = r a+b . + r a+b , where b + b' = a

3

- series: E - r J+ i + rj+i = 3T

- series: E = r 4+g + r i+i = 4T
4

Digit

series: E = r^s + r M = 5T
series: E = r^ 6 -f r«+i = 6T

Google

Original from
UNIVERSITY OF MICHIGAN

190 THEORY OF REGULARITY AND COORDINATION

7

- series: E = r 7+4 + r 7+J = 7T
o

- series: E = rg+s + r 8+s = 8T
o

9

-series: E = r 9 +j + r»+ 4 = 9T

~ series: E = rio+7 + rio+j = 10T

11 . _ _
-- series: E = r n+8 + ru+s = 11T

|^ series: E = r 1!+ 7 + rn+t = 12T
13

— series: E = ri S+7 + ru+« = 13T

7^ series: E = r u +» + ru+6 - 14T
14

|| series: E = r iM + r iM = 1ST

~ series: E = ri«+9 + r 16+7 = 16T
lo

17

— series: E = ri 7+ » + r n+ g = 17T
|| series: E = ris+n + r 18+7 = 18T

1 o

19

— series: E = ri» +u > + r u +» = 19T
20

— series: E = r 20 +n + r J0+ 9 = 20T

Second Form: in composing the resultants of interference of two synchronized
biners, select two non-identical major generators (a and a') which are the succes-
sive terms of one summation-series, and whose sum equals the value of the
determinant of the series (and, hence, the value of T); then select the minor
generators (b), which are identical and whose value equals the difference be-
tween a and a'; the value of a is greater than that of a', or vice-versa.

r\r\Ci\f> Original from

Digitized by ^UUJjk UNIVERSITY OF MICHIGAN

BALANCE, UNSTABLE EQUILIBRIUM

191

Formula:

E' = r a+b + r a - +b and/or

E" = r a 'j. b + r a+b , where a' — a = b

The value of coefficients of r,.^. b and r a+b equals the value of T divided by 2;
each half becomes the coefficient of recurrence of either r; when T is an odd

T + 1

number, the value of the first coefficient is » and the value of the second

T - 1 2

• or vice versa.

2

Complete formula for T = 2n:

T T

E = - r a+b + - r a - +b and

T T

E — 2 r a'-i-b + r a+b

Complete formula for T = 2n + 1 :

v , T + 1 , T - 1

E = — ^ — ra+b 2 — ' V ' ! ' b an

T + l T + 1

E = — - — r a - +b H - — r a ^. b and

T - 1 T + l

E = — ~ — r a+b H — r a - +b and

T - 1 T + 1

E = — - — r a . +b -\ — r a+b

Examples:

8 .

- series: E = 4r5+ 2 + 4r 3 +2 = 51 +31
o

9

- series: E = + 4r 4 + s = 5T + 4T
10

— series: E = 5r 7+2 + 5r J+2 = 7T + 3T
12

— series: E = 6r 7 ^2 + 6r*+ 2 = 7T + 5T

rv M nr\cs\£> Original from

Digitized by ^UU^IC UNIVERSITY OF MICHIGAN

THEORY OF REGULARITY AND COORDINATION

— series: E = 7r 7+6 + 6rj+ 8 = 7T + 6T
13

77 series: E = 7r» +4 + 7r 6 + 4 = 9T + 5T
14

^ series: E = 8r 8 +s + 7r 7 +s = 8T + 7T

|— series: E = 8r g +j + 7r 7+6 = 8T -f 7T

jf series: E = 8r 9+2 + 8r 7+2 = 9T + 7T
16

7^ series: E = 8r 9+4 + 8r 7+4 = 9T + 7T
16

77 series: E = 8r 9+6 + 8r 7+i = 9T + 7T
16

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

CHAPTER 7

RA TIO A ND RA T ION A LIZ A TION

A. Ratio. Rational. Relation. Relational

"DATIONAL behavior — behavior according to a ratio. Rational composition —
■"■^composition based on a ratio. Rational thinking — thinking in terms of ratios.
When the ratio is established, involution (power-differentiation) takes its course.
Cutting a portion of space by simple (monomial) or complex (polynomial) periodic
motion establishes an area. Thus, enclosing an unbounded space in a rational
boundary ipso facto introduces regulations that are the inherent laws within the
boundary. The act of limiting converts potentiality into a tendency (intent).

This process can be defined as ratio-realization of space. Viewed in this way,
the inscribing of a structure (trajectory) in a boundary* which structure was
originally evolved in an unbounded space, corresponds (is equivalent) to ration-
alization of structure.

These are the two fundamental procedures, mutually compensating each other.
In terms of logic, one corresponds to relativity and the other to quanta, as the
first works from an enclosed (bounded) all-inclusive whole, while the second
operates on a unit (st). The combination of the two bases of departure establishes
a unified system of interacting tendencies of the unit and the whole.

B. Rationalization of the Second Order
(Rationalization of a ratio).

Introduction of a new or of an identical ratio rationalizes a given ratio, thus
introducing symmetry into a given ratio. If a given ratio is 2, and its extensional
equivalent is 2, then the introduction of a 2:1 ratio gives ■§• -f likewise, 1 -5- 2
ratio yields £ + § . The combination or interference of the two yields ^ as a
unit of symmetric distribution.

To demonstrate that there is symmetric correspondence between the extension
and the ratio, let us take a frequency 2-5-1 ratio. In a ratio of two frequencies,
the extraction of the root (evolution) corresponds to division as applied to quanti-
tative extension. Then, taking the ratio 2, which is the equivalent of 2-5-1, we
extract \/2. This makes one third of the ratio value, as y/T} = 2. Therefore
yfl is the equivalent of one third in an extension. Or to put it mathematically:
IogJ^v/2 = 1, log 2fV¥ = 2, log 2jT\/T> = 3.

Further evolution is achieved by squaring the exponent of the radical. Thus,
to establish a new 2 -5- 1 ratio within each of the -s/l, it is necessary to extract
y/2. Then, f/V gives the equivalent of $ of the extensional value, i.e., \/2*

io» Original from

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194

THEORY OF REGULARITY AND COORDINATION

As an illustration, we can take the 2 -hl_ratio of sound frequencies. This forms
an octave. By extracting the \/2 and \/2 2 , we obtain the following three sym-
metric intervals between the frequencies:

(1) from 1 to y/2 '

(2) from y/2 to y/V

(3) from y/2 1 to 2

Assuming 1 to be 256 cycles, we obtain the following values for the symmetric
points:

(1) ^512

(2) ^sTT 1

(3) ^512» = 512

The musical names of these points are: c, e, g#, c'.

The limit frequency between 1 and \/2 in the temperament of \/2 is ir - =

V2

\/2. Thus \/2 becomes the limit of frequency units in the symmetry y/2.
As \/2 = 3\/2, we can establish a binomial ratio within this limit. In our
original reasoning, we established the symmetry of three intervals to the octave
as the series, in which we have developed the 2-5-1 ratio. Now we assume the
coefficient 3 of the 3v / 2 to belong to §■ series as well. Thus we establish a bi-
nomial symmetry of 2-4-1 within \/2. It appears in the following form : 2v^2 -f
y/2, which in logarithms to the base \/2 becomes 2 + 1. As 1 is the equiv-
alent of a musical semitone, we obtain the following pitch-scale, which carries
out 2-j-l ratio from all three symmetric points:

c — d — eb — et] — f# — g — g# — a# — b — (c).

Similar ratios can be carried out within assigned ratios of wave-length for the
projection of spectral colors. Color-scales may be constructed in any form of
symmetry within the assigned range (ratio of wave-length) and from any sym-
metric points.

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

RATIO AND RATIONALIZATION

195

C. Rationalization of a Rectangle

The rationalization of a rectangle consists of the process of subjecting the rect-
angle to the tendency of its own ratio (i.e., the ratio of two sides of the rectangle:
a :b) within a new ratio. The new ratio must be one of the simplest. In the following
discussion, the rectangle itself may have any ratio, but the new (assumed) ratio
is 2, and will be used as a constant. The ratio of 2 represents an octave and is a
selective operation of the most general kind. It is applicable to frequencies, wave-
lengths, etc. ; now the application will be extended to a rectangle.

The rationalization of a rectangle has the purpose of extending the original
ratio to half of the sum of two sides of the rectangle and to the remaining portion
of the longer side. After such an operation has been applied to the two above-
mentioned segments, each of the two segments becomes subdivided into two
shorter segments which are related to each other as a:b. Thus we acquire four
extensions on the longer side of the rectangle.

Assuming that the original rectangle has sides a and b, and assuming further
that the sum of the segments of proportionate subdivision xi, x 1( xj and x 4 equal a,
we find that the above segments bear the following relation to each other:

a
b

Xl
Xj

X»
X 4

By dropping perpendiculars from the points between xi and xj, xj and x», and xi
and x«, we isolate four areas: Ai, A 2 , At, A 4 . These areas bear the same relation to
one another as the segments xi, xj, x$ and x 4 :

A,

A,

a

A,

A 4

b

a

Xl

Xi

X,

x 4

A,

A s

A,

A 4

According to our conditions, xi + xj =

a + b

w , a + b , 2a + 2b a+b 2b 2a+2b-a-b-2b
Then,x,+x 4 =(a + b) - b = — y = -

a — b . . .. . . a + b a — b

= Thus, side a is subdivided into two extensions: — - — and — - — •

2 2 2

Any area of four sides, that is, a rectangle, can be expressed as the following

trajectory:

rv ••■ wh (^f\r\cs\t> Original from

Digitized by VjUUVI^ UNIVERSITY OF MICHIGAN

196

THEORY OF REGULARITY AND COORDINATION

A4 - 2 (sat 180° + 90* + sbt 180° + 90°). In other words, the boundary
of a four-sided rectangular area equals two times the rectilinear extension whose
period is at, turning under 90° into another rectilinear extension whose period is
bt. Disregarding 180° as a general constant of rectilinear extension and 90° as a
rectangular constant, we can express the boundary of any specified rectangle as:

A4 — 2(at + bt;, where a and b are the periods producing a 90° angle, and
therefore — the ratio of the two sides of A4.

Assuming the sum of two sides of a rectangle to be: A4 = a+b, we shall

a+b

designate such a sum as the determinant of the series. For example: A4 =

a + b

2(2t + 1) expresses the rectangle whose area is 2 X 1. This particular rectangle be-
longs to \ series, as its determinant is: 2 + 1 ■» 3.

I. Formulae

(a) Extensions of the Major Term.

a + b

a

a 1 + ab

a (a + b)

a

2

a + b

2a + 2b

2(a + b)

= 2

a+b

b

ab + b l

b(a + b)

_ b

2

a+b

2a + 2b

2(a + b)

~ 2

a - b

a

a J — ab

a(a - b)

2

a+b

2a + 2b

2(a + b)

a - b

b

ab - b l

b(a - b)

2

a+b

2a + 2b

2(a + b)

(b) Ratios of the Extensions.

Xi

a* + ab

ab + b J

a 1 + ab

a

Xi

2a + 2b

+ 2a + 2b

ab + b 1

= b

X|

a* — ab

ab - b J

a 2 — ab

a(a

-b)

a

X 4

2a + 2b

+ 2a + 2b

ab - b 1

b(a

-b)

= b

Xl

a 1 + ab

a J — ab

a 1 + ab

a(a

+ b)

a

+ b

X|

2a + 2b

+ 2a + 2b

a 1 — ab

a (a

- b)

a

- b

Xj

ab + b'

ab - b 1

ab + b l

b(a + b)

a

+ b

X 4

2a + 2b

+ 2a + 2b

ab - b 2

b(a

-b)

a

- b

Xi

X| a

Xj Xj

a + b

Xi

" x~« " b 1

X| x«

a - b

Original from
UNIVERSITY OF MICHIGAN

RATIO AND RATIONALIZATION

197

(c) Minor Areas (Subareas) of a Rectangle.
A, =

A,=
A,=
A< =

a 1 + ab

h

a 2 b + ab 1

ab(a + b) ab

2a + 2b

U

2a + 2b

2(a + b) 2

ab + b 2

b

ab 1 + b»

b 2 (a 4- b) b 1

2a + 2b

2a + 2b

2(a -f b) 2

a 1 — ab

b

a 2 b - ab 1

ab(a - b)

2a + 2b

2a + 2b

2(a + b)

ab - b 2

b

ab 1 - b*

b l (a - b)

2a + 2b

2a + 2b

2(a + b)

(d) Ratios of the Minor Areas of a Rectangle.

Ai

a 2 b + ab 1

ab 2 + b'

a J b + ab 1 ab(a + b)

ab _

a

A 2

2a + 2b

2a + 2b

ab 1 + b» b l (a + b)

b 2 ~

b

A,

a l b - ab 1

ab 2 - b»

a J b — ab 2 ab(a — b)

ab _

a

A 4

2a + 2b

' 2a + 2b

ab 2 - b» b 2 (a - b)

b 2 ~

b

A t

a 2 b + ab 5

' a 2 b - ab 1

a 2 b + ab 2 ab(a + b)

a + b

A,

2a + 2b

2a + 2b

a 2 b — ab 2 ab(a — b)

a —

b

A,

ab 1 + b*

ab 1 - b»

ab 2 + b J b 2 (a + b) a

+ b

A 4

2a + 2b

2a + 2b

ab 2 - b» b 2 (a - b) a

- b

A,

A 3 a

Ai A t a

+ b

A,

A 4 b'

As A 4 a

- b

(e) Major Areas (Subareas) of a Rectangle.

+ a a ' b+ab2 , ab 2 4-b» _ a 2 b-|-2ab 2 +b' _ (ab+b 2 )(a+b) _ ab+b 2
2 ~ 2a+2b 2a+2b~ 2a+2b ~ 2(a+b) ~ 2

a 2 b-ab 2 , a 2 b-b» a 2 b-ab 2 +ab 2 -b* a 2 b-b J b(a 2 -b 2 )

A,+A 4 =

2a+2b 2a + 2b 2a+2b 2a + 2b 2(a+b)

b(a+b)(a-b) _ b(a-b) _ ab-b t
2(a+b) 2 ~ 2

Ratio of the Major Areas of a Rectangle.

A t + A 2 _ ab + b 2 , ab - b 2 _ ab 4-b 2 _ b(a + b) _ a+b

198

THEORY OF REGULARITY AND COORDINATION

ffl Summarv

\* J WW-' ""111 JF *

xi Ai x*

A*

Xj As x<

" Ai

xi _ Ai _ xj

_ A_ t

xi Ai Xi

A»

Ai Ai a

+ b

Ai A 4 a

- b

2X1 rectangle

2 + 1

= 3

3 3

3

" 3

a
b

2. Application

Ratio of two sides = 2

3 3
3*2 ~ 6

3-2 6
6-6 ™ 36

;A,

1-2

~ 6-6

2

" 36

6 2

1

36 ' 36

6

~ 2

_ 3
~ 1

Ai

Ratio of two sub-areas (Aj and A») = 3

Digiti

+-

-

2

-i

Figure 1. Rationalization of rectangle 2X1.

Original from
UNIVERSITY OF MICHIGAN

oogle

RATIO AND RATIONALIZATION 199

3X1 rectangle

3 - + l - = *-

4 4 4

_4_ _ 4
4-2 ~ 8

4-2 _ _8 ^2 4

1 " 8-8 " 64 ; " 8 8 " 64

A, _ _8 ^ _4_ _ 8-64 = 2
Ai " 64 * 64 ~ 4*64 ~~

3X1 RATIO=l»gi»Ai

Xy X*

Figure 2. Rationalization of rectangle 3X1.

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

202

THEORY OF REGULARITY AND COORDINATION

3X2 rectangle

3 2 _ 5
5 5 = 5

_5_ 5^
5-2 " 10

4 JO A _ 1-4 _4_

10-10 100 1010
A, 20 4 20 100

A t 100 100 4 100

- 5

3*1 *ATlO»5»fj*£*

lii

Figure 4. Rationalization of rectangle 3X2.

Digitized by GoOgle

RATIO AND RATIONALIZATION

5X2 rectangle

7-2 14

_ 7 ' 4 _ _2JL A _ 3 4 _ _12_
1 " 14-14 ~ 196 : ' ~ 14-14 ~ 196

Ai _ 28^ ^ 12_ _ 28-196 _ 7
A, ~ 196 ' 196 ~ 12-196 ~ 3

7X3 rectangle

_7_ 3 _ 10
10 10 ~ 10

10 10
10-2 ~ 20

10-6 60 6-4 24
1 ~ 20-20 ~ 400 ' 20-20 ~ 400

Ai _ 60 _ 5
A, ~ 24 ~ 2

4X3 rectangle

A,

4 3 _ 7
7 7 ~ 7

7

7

7-2 =

: 14

7-6

__ 142

7

14-14

~ 196 1

14

1-6

6

A,

14*14

" 196 :

A,

Digil

14

oogle

Original from
UNIVERSITY OF MICHIGAN

204 THEORY OF REGULARITY AND COORDINATION

5X3 rectangle

5 3 _ 8
8 8 " 8

_8_ ^ 8^
8-2 ™ 16

A,

5X4 rectangle

8-6

: 48

8

6

_ _!

16-16

~ 256 :

16

16

= 16

2-6

12

A,

48

- 4

16-16

~ 256 1

A,

~ 12

5 4 = 9
9 9 ~ 9

_9_ _ _9
9-2 " 18

A, =
A, =

9-8

_ 72

9

8 _ 1

18-18

" 324'

18

18 ~ 18

1-8

8

A,

72-324

18-18

~ 324 :

A,

324-8

6X5 rectangle

_i . A _ 11
11 11 ~ 11

11

_ 11

11-2

~ 22

11-10

110 11

10 _ J_

22-22

~ 484 1 22

22 ~ 22

1-10

10 Ax

110-484

22-22

" 484 ! A,

484-10

Ax =

A, = — — = — : T- = TTTTT^ =H

rv - ■ r\r\cs\f> Original from

Digitized by VjUUJJU. UNIVERSITY OF MICHIGAN

RATIO AND RATIONALIZATION

7X2 rectangle

7 2 _ 9
9 + 9 ~ 9

9

9

9-2 =

= 18

9-4

36 9

4

5

18-18

~ 324 : 18

~ 18

~ 18

5-4

20 A,

36

9

18-18

~ 324 : A,

~ 20

~ 5

7X3 rectangle

7_ 3^ _ 10
10 + 10 ~ 10

10 10

A, =
A, =

10-2 20

10-6 60 10 6 4

20-20 400' 20 20 20

4-6 _ _24_ Ai _ 60 _ 10 _ 5
20-20 ~ 400 ! A a ~ 24 ~ 4 ~ 2

7X5 rectangle

7_ 5 12
12 + 12 ~ 12

12 12

A, «

12-2 24

12-10 120 12 10

24-24 576' 24 24 24

2-10 20 Ai 120
* ~ 24-24 " 576 : A, " 20

Digitized byGoOgle

Original from
UNIVERSITY OF MICHIGAN

206 THEORY OF REGULARITY AND COORDINATION

7X6 rectangle

]_ 6 _ 13
13 13 ~ 13

13 _ 13
13-2 ~ 26

*

13-12 _ 156 13 _ 12 _ J_
26-26 ~ 676' 26 26 ~ 26

1-12 12 A, 156
26-26 ~ 26 : A, ~ 12

8X3 rectangle

A A 11
11 + 11 ~ 11

11 _ 11
11-2 ~ 22

_ AlA _ H _ 6 A

1 ~ 22-22 ~ 484 : 22 22 ~ 22

_ 5-6 30_ Ai _ 66 _ 11
3 ~ 22-22 ~ 484 : A, ~ 30 ~ 5

8X5 rectangle

A, =

5 _ 13
13 ~ 13

13

_ 13

13-2

~ 26

13-10
26-26

_ 130 13
~ 676' 26

10 _
~ 26 ~

3-10

30 A,

130

26-26

~ 676 ; A,

30

3^
26

Digitized byCiOOgle

13
3

Original from
UNIVERSITY OF MICHIGAN

RATIO AND RATIONALIZATION

8X7 rectangle

— 4- — - —
15 15 ~ 15

15 _ 15
15-2 ~ 30

15-14 _ 210 15 _ 14 _ J_
30-30 ~ 900' 30 30 ~ 30

1-14 _ U_ Ai 210 30
30-30 ~ 900' A, ~ 14 ~ 2

n , , Ai 9 + 2 11

9X2rectangle:-=— 2 = y

11

— series

„ , , A, 9 + 4 13

9X4 rectangle : — = = —

Ai 9 — 4 5

13

— series
13

, A, 9 + 5 14 7

9 X 5 rectangle: -= — = - = -

14

— series
14

n n , Ai 9 + 7 16 „

9 X 7 rectangle: — = — - = - = 2

16

— series
16

A 9 + 8

9X8 rectangle: — 1 = = 17

A, 9 — 8

17

— series
17

rv in _j i» r\rscs\f> Original from

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208 THEORY OF REGULARITY AND COORDINATION

, A t 10 + 3 13
10 X 3 rectangle: - = — = y

13

— senes

1 o

, A, 10 + 7 17
10 X 7 rectangle: - = — = -

17 .
— senes
17

, A, 10 + 9 <rt
10 X 9 rectangle: — = — — - = 19
Aj 10 — "

19 .
-senes

, A, 11 + 2 13
11 X 2 rectangle: - = — = -

13

— series
13

, A, 11+3 14 7
11 X3 rectangle: -=— - y - j

14

— series
14

, A, 11+4 15

11X4 rectangle: — = — = —

A3 11—4 7

15

— series

. Aj 11 + 5 16 8
11 X 5 rectangle: - = rrr ^ = - =-

16

— senes
16

, A, 11+6 17

11X6 rectangle : — = — = —

A3 11—6 5

17 .
— series
17

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

11 X 7 rectangle

RATIO AND RATIONALIZATION
A, 11 + 7 18 9

As 11-7 4
18

— series
18

, Aj 11 + 8 19
•I* 8 rectangle:-- — = J

19 .

— series
19

, Ai 11 +9 20 in
11X9 rectangle: - = ^fZ^ = J = 10

20 .
20 SCneS

, Ax 11 + 10 21
llXlOrectangle:-= IT - ro = T = 21

21

— series
21

, A > 12 + 5 17
12 X 5 rectangle: - = — = y

17

— series
17

, Ai 12 + 7 19
12 X 7 rectangle: -= — = y

19

— series

rv -■ _j l nr\Ci\(> Original from

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210 THEORY OF REGULARITY AND COORDINATION

J. Charts.

Series : a + b

Rectangle : A4

Ratio of A] : A s

O

2 X 1

o

3

4

3 X 1

2

A
t

5

c

3X2

5

5

4 X 1

?
>j

6

5 X 1

3

D

o
z

7
7

4X3

7

5X2

7

3

6 X 1

7
c

8

8

5X3

4

7 X 1

4

3

9
9

5X4

9

7 X 2

9

5

8 V 1

o A, I

9

7

10

7 X 3

5

10

2

9 X 1

5

Digitized by GoOgle

RATIO AND RATIONALIZATION

211

Series : a -f- b

Rectangle : A4

Ratio of Ai : A 3

11
11

6X5

11

7X4

11

3

8X3

1 1

5

( ) X 2

11

7

10 X 1

it

9

12
12

. 7X5

12

5

11 X 1

6
5

13
13

7 X 6

13

8X5

13

3

9X4

13

5

10 X 3

13

7

11X2

13
9

12 X 1

13

14
14

9X5

7
2

11X3'

7
4

1 13 X 1

7
6

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Original from
UNIVERSITY OF MICHIGAN

212

THEORY OF REGULARITY AND COORDINATION

Series : a -f b

Rectangle : A4

Ratio of Ai : A 8

15
15

8 X 7

15

11 X 4

15

X \J

7

13 X 2

IS

TT

14 X 1

15

16
16

9 X 7

2

11 X 5

8
3

13 X 3

8
5

15 X 1

8
7

17
17

9X8

17

10 X 7

17

T

11 X 6

17

7

12 X 5

17

y

13 X 4

17

~9~

14 X 3

1 7

X 1

11

15 X 2

17
13

16 X 1

17
15

Digili

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Original from
UNIVERSITY OF MICHIGAN

RATIO AND RATIONALIZATION

213

Series : a + b

Rectangle : A4

Ratio of A t : A 3

18
18

11 X 7

9
2

13 X 5

9
4

17 X 1

n

V

8

19
19

10 X 9

19

11X8

19
3

12 X 7

19

5

13 X 6

19

7

14 X 5

19

15 X 4

19

TT

16 X 3

19
13

17 X 2

19
15

18 X 1

1 o

20
20

11X9

10

13 X 7

10
3

17 X 3

10

7

19 X 1

10
9

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Original from
UNIVERSITY OF MICHIGAN

214 THEORY OF REGULARITY AND COORDINATION

Series : a + b

Rectangle : A4

Ratio of Ai : A 8

21
21

11 X 10

21

13 X 8

21
5

16 X 5

21
11

17 X 4

21
13

19 X 2

21
17

20 X 1

21
19

D. Ratios of the Rational Continuum 1

5-5-3 7 + 5 9 + 7 11-4-9 13-5-11 15 + 13 17 + 15
2-f-l 3 + 2 4 + 3 .5 + 4 6 + 5 7 + 6 8 + 7

194-17 21-1-19 234-21 25 + 23 27-5-25

94-8 104-9 114-10 124-11 13412 144-13

294-27 314-29 334-31 354-33 374-35

154-14 164-15 174-16 184-17 194-18

394-37 414-39 434-41 454-43 47 4-45

204-19 214-20 22 + 21 234-22 244-23

494-47 514-49 534-51 55 + 53 574-55

254-24 264-25 274-26 284-27 294-28

594-57 614-59 63 + 61 654-63 67 4-65

304-29 31+30 324-31 334-32 34 4-33

694-67 714-69 734-71 754-73 774-75

35+34 36 + 35 37 + 36 38 + 37 39 + 38

79 + 77 81 + 79 83 + 81 85 + 83 87 + 85

40 + 39 41+40 42 + 41 43 + 42 44 + 43

89 + 87 91+89 93 + 91 95 + 93

45+44 46 + 45 47 + 46 48 + 47.

iThe reader is referred to Appendix B. which
contains the relative dimensions resulting from
these ratios. (Ed.)

Digitized byL^OOglC

UNIVERSITY OF MICHIGAN

CHAPTER 8

POSITIONAL ROTATION

POSITIONAL rotation (p. r.) is equivalent to circular permutation with the im-
plication of the concept of sequence. The use of this term signifies a sequent
group evolved by means of circular permutation: for example, positional rotation
of coordinate phases of music: melody evolved by means of p. r. ; harmony evolved
by means of p.r. ; and correlated melodies (counterpoint) evolved by means of
p. r.«

In design: composition of superimposed images obtained by means of p.r.;
kinetic sequence of images (cinema) in p.r. ; and color sequence obtained through
p.r. In the coordination of the groups of components, such as scores of homo-
geneous or heterogeneous arts, p.r. can control simultaneity (ordinate phases) and
continuity (abscissa phases). This device permits variation of a pre-sct coordi-
nated group automatically, both in simultaneity and in continuity, either in the
positive (O ()) or in the negative (O 0) direction. Two-dimensional (two
coordinate) positional rotation may be expressed in terms of the phasic and the
directional relations.

A. Dimensionality of Positional Rotation

Positional rotation can be defined as two-dimensional (x, y) and two direc-
tional (positive: O x, () y, and negative: O x, y)- Each phase of positional
rotation is defined by the dimensional and directional relations of x and y. Ex-
pressing a simultaneous structural group as S, its units — as s, general time
period as T and its duration units as t, we may elaborate the four fundamental
forms of positional rotation.

(1) SmsOTntC; (2) SmsOTntC;

(3) Sms C Tnt Q; (4) Sms Q Tnt Q.

Here m and n represent the phasic values. As C and () represent the posi-
tive values of t and s respectively, and C an d represent the negative values
respectively, any desired st phase of positional rotation may be computed
directly. The definition of structure (S) in its original position is as follows:

S = ms nt . The limit position of m is Sm, and the limit position of t is t„.

The change of position S„ in a positive phase results in S a position. The
change of position t in a positive phase results in t a position. The change of
position S Q in b negative phases results in S_ b position. The change of position
tj in b negative phases results in t_ b position.

^hcse processes are described in detail in The
Schillingtr System of Musical Composition. (Ed.)

f\f%Cs\f>7iK Original from

by VjUUgl^zii> UNIVERSITY OF MICHIGAN

216 THEORY OF REGULARITY AND COORDINATION

mn

ST ST = mynx

oo

S()TC S = ms

T = nt

S () T O ST = msnt

S {) T Q The original position:

S T = ms nt o
S()TC StT, -s,t,

S k T_ b = s k t_ b

mn

ST = S t , Sjt , S tj, Sjti, S2t ,
oo

S2ti, S c t2, Sit 2 , S2t2, . . .
S m t<» S m ti, S m t2, . . .
S t n i Sit n , S 2 t n ,
S n t n -

46

ST = S t , Sjt , Sjti, . . . S^, . . . S2t 6
oo

If a given phasic position is S s t2, the addition of 2S — 2t gives: Sjti +
+ (2S - 2t) «= S 5 t

Let the structure be :

48

ST

oo

(1) S t + (5s - 3t) = S^

(2) S t + (-2S + lit) = S,t s

(3) S t + (3S + 5t) = S s t 6

(4) S t o + (3t + t + 2t + 2t + t + 3t) = S t 4

(5) S Q t + (3t - t + 2t - 2t + t -3t) = S D t

(6) S t + (3S - t + 2S - 2t + S - 3t) = S,t,

f^ru^nl^ Original from

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POSITIONAL ROTATION

217

Let the structure be: 66

ST

oo

then : S D = S», t = t 6

(1) S t + (2S-t+s-2t) =S,t 8

(2) S t + (3S - t + 2S - 2T + S - 3t) = S t,

The addition of a resultant or of any group with a compensating form of
symmetry, at the end of the group, restores the latter to its original position
when the signs of consecutive terms alternate, and the number of terms is even.

It is possible to compute directly any phase either from zero phase or from
any other phase for both s and t.

Example: mn

(1) ^ + (5S + 2t - S - 3t + 2S + t) = S t c + (6S + Ot) = S,t

(2) S 2 t 2 + (5S + 2t - S + 3t + 2S + t) = S 2 t 2 + (6S + Ot) = S 8 t 2

In general:
if t = tn

t + nt = nt — nt = t
ti + nt = nt - 1 = ti
tj + nt = nt - 2 = t 2

t = t«
tp + t - ti
t + 2t = t,

t + 6t = t

t + 7t - t + (7t - 6t) = t,

t + 8t = t + (8t - 6t) = t,

(^ru^filr* Original from

Digilized by ^UU^IL UNIVERSITY OF MICHIGAN

ST

oo

6
T

o

218

THEORY OF REGULARITY AND COORDINATION

If m > n, then:

t + mt = t + (mt — nt) = tm — n
If m > 2n, then:

t + mt = t n + (mt - 2nt) = tm — 2n
If m > pn, then :

t D + mt = t -f (mt — pnt) = tm — pn

to

+ 1 = t

i

to

+ 2t =

t 2

to

+ 3t =

ts = t

to

+ 4t =

(4t - 3t) =

ti

to

+ 5t =

(St -3t ) =

t 2

to

+ 6t =

(6t - 2-3t)

= t

to

+ 7t =

(7t - 6t) =

ti

to

+ 8t =

(8t - 6t) =

t 2

to

+ 9t =

(9t - 3-3t)

= t

to

+ lOt =

= (lOt - 9t)

= tl

As temporal units (t) of the structural temporal group (T) move back and
forth, controlled by definite positive and negative coefficients, physical time
(during which perception takes place) evolves in one direction. Thus, each
temporal sub-group of the entire T corresponds to t with some coefficient. We
shall indicate it as mt, nt, pt, ... The positional zero of any subgroup therefore
corresponds to P, which is the term of physical time (the positive non-reversible
time). If Pi is the original term of physical time, P 2 is the second term of physical
time, etc.; then: Pi = mt, P 2 = nt, P 3 = pt, ... All these considerations hold
true when t jumps back and forth with any coefficient, but never actually moves
backwards. Positional rotation presupposes phasic variations of T or of its sub-
groups mt, nt, pt, ... where T, mt, nt, pt, ... are treated as discontinuity.
In order to produce continuity of phasic rotation, it is necessary to introduce the
dense set from t D to ti, from ti to t 2 , etc., and to use such dense sets in the positive
as well as in the negative direction.

3

Example: T

o

Digit

Original from
UNIVERSITY OF MICHIGAN

CHAPTER 9

SYMMETRY

A. Symmetric Parallelisms

(Configurational Identities)
3

(1) - series

(a) Six-pointed star (Hebrew mogen dovid) represents binomial periodicity
of the $ series: 2 + 1 = Z 120°+ Z60°; 6(120° + 60°) in alternating
direction, usually represented as two superimposed equilateral tri-
angles; in combination with the cross (-j-f) series: 4(5+5+2) in constant
direction under Z 90° and heart (bifold symmetry) ($ series) used by
the Franciscan order. It is of importance that 120° + 60° produce
180°, i.e., an infinite rectilinear extension.

(b) All members of $ series participate in the temporal configurations of
Catholic liturgical music, especially in the 14th to the 16th centuries.
This was an intentional selection of the symbol of Divine Trinity.
This form influenced all the subsequent forms of European music writ-
ten in |- time, which forms evolved later in the 18th and the 19th cen-
turies into hexagonal forms of symmetry ($ series).

(c) A secondary selective system evolved by the Arabs in the 7th century
A.D.: a pitch-scale known as "zer ef kend." (literally: string of pearls).
This scale was conceived as an alternation of a large step (bead) and
a small step (bead) :

Figure J. String of pearls. "Zer ef kend."

Original from

219 UNIVERSITY OF MICHIGAN

220 THEORY OF REGULARITY AND COORDINATION

Zcr cf kcnd approximates with sufficient precision the alternation of a
whole tone and a half tone of the \/2 temperament:

c-d-eb-f-ftf-gtf-a-b-c 1
2 + 1+ 2 + 1+ 2 + 1 + 2+1

It means that the binomial of the f series, after performing four cycles
(as in the case of beads), closes in an octave, i.e., in -f- ratio: 4(2 + 1) = 12,
which links it with the hybrid yf series, i.e.,

3 4 _ 12
3 ' 4 ~ 12*

i

The latter, in turn, links it with the "blues", which is partly hybrid,
partly pure \% series.

(d) Temporal configurations of the music of Oklahoma Indians:

1 1 3

3... 9

9 3 3

(2) - series
5

(a) The natural pentagonal and pentaclic developments: sea urchin, star-
fish. The starfish is r ( &+2), where the \ series is the primary genetic factor
(as the starfish at first represents bifoldedness without signs of pentag-
onality) and the \ series is the secondary genetic factor (possessing the
pentagonal tendency).

r (6 +j) = 2+2 + 1+1+2+2, and refers to the sequence of angles:

Figure 2. Sequence of angles in pentagonal form.

iThe numbers given here and elsewhere to de-
note pitch scales refer to semitones. The number
2 denotes two semitones or a whole tone (c — d) .

Three (3) denotes three semitones (c — eb). etc.
(Ed.)

i„;,i^hu( nnnlp Original from

igitizeti by ^.UUgl v. UNIVERSITY OF MICHIGAN

SYMMETRY

221

(b) The primary selective system of Javanese tuning:

y/2 ("slendro").

(c) The pentameter of verse coupled with the ^ temporal series of music of
Asia and European Russia.

(d) The Caucasian rug- patterns, where the width of partial rectangles is
arranged as the sequence of terms of !"(»+»), i.e., 3+2 + 1+3 + 1+2+3.

(e) Distribution of the visible spectral circle into S uniform sectors : -tywP

(3) g s*"«

(a) Hexagonal symmetry of honeycombs.

(b) Primary selective system of tuning: \/2, i.e., the "whole-tone scale".

(c) Adaptations of hexagons in tiles, floors (wood inlay) in Persia, Rome, etc.

Figure 3. Practical form off series {star), f series (heart), and f§ series (cross).

rv in _j i» r\rscs\{> Original from

Digitized by V^UUgl^ UNIVERSITY OF MICHIGAN

222

THEORY OF REGULARITY AND COORDINATION

Courtesy National Gtogtapkit Magazine and Buffalo Museum of Science

Figure 4. Snowflakes are all based on $ series.

j Unginal trom

Digitized by UNIVERSITY OF MICHIGAN

SYMMETRY 223

^ J

B. Esthetic Evaluation on the Basis of Symmetry

Beauty as a form of Regularity or Symmetry.

A perfect case of symmetry is the Arabian pitch-scale "Zer ef kend" (String
of Pearls) : 4(2 + 1 ) . I ts derivative (d i) is : 4( 1 +2) .

Among European scales (ecclesiastic modes), we find the following struc-
tures: 2

(2 + 2 + 1) + (2 + 2 + 2)

Nat. major

do

(2 + 1 + 2) + (2 + 2 + 1)

Dorian

d,

(1 + 2 + 2) + (2 + 1 + 2)

Phrygian

d 2

(2 +2 + 2) + (1 + 2 +2)

Lydian

d 3

(2 + 2 + 1) + (2 + 2 + 1)

Mixolydian

d 4

(2 + 1 + 2) + (2 + 1 + 2)

Aeolian

d 6

(1 + 2 + 2) + (1 + 2 + 2)

Locrian

d 6

Esthetically, grade A are: d 4 , d 6 , d B
" B " d,,d s

" C " dfl, d,

Grade A are symmetric

" B " modified symmetric

" C " assy metric

It is interesting to note that in Russian folk music, only d 4 and d B (both
symmetric) are commonly used, d, being an exception. In ancient Greek music d4
(which was known as Mixolydian) was dedicated to the Sun, and was therefore
regarded as the royal scale. All other scales were dedicated to different planets,
de is symmetric.

The fundamental structure of the so-called "Chinese pentatonic" scale is:
2 + 2+3-1-2 or 2+3+2 + 2. It is interesting to note that 2+2 + 2=6 and the re-
maining interval is. 3, thus forming a § = y ratio between the sums of the two
kinds of intervals employed. Perfect bifold symmetry appears in d& of the origi-
nal scale: a— c — d— e — g, i.e., 3+2+2+3, a frequently used scale.

One of the prominent Javanese scales is constructed downward: a— g— e
— d — b, i.e., 2+3+2+3. This scale is also prominent in Madagascar. 3 It is known

*The symbol d is used to denote displacement or how the various modal scales may be derived by
circular permutation, do is the zero displacement
scale; di is the first displacement scale that may
be derived from it by circular permutation; dj is
the second displacement, etc. Schillinger shows

Original from
UNIVERSITY OF MICHIGAN

displacement from natural major. (Ed.)
'Compare Ravel's Chansons Madecasses.

Digit

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224 THEORY OF REGULARITY AND COORDINATION

as "Slendro". The other fundamental Javanese scale is "Pelog" (y/2). Reading
downward: f — e — c — b, i.e., 1 + 4+ 1, which is the central trinomial of the
$ series. The Balinese scale "Selenders" (derives from the Javanese \/2 tuning
Slendro): (3+2+3+2) + (2+3+2+3) +2 (reading downward). The Balinese
dance "Djanger" is based on the scale: (2 + 1+4) + (2 + 1+4), reading down-
ward: d— c — b— g— f — e— c. Persian popular songs are based on (reat ing down-
ward): g-f-e— d-c-b-a, i.e., (2 + 1+2) + (2 + 1+2), a scale identical with
the Aeolian mode of Glareanus.

Our pitch discrimination is conditioned by y/2, our time discrimination
(in music and movement) by 2.

C. Rectangular Symmetry of Extensions in Serial Development

(5+1)+ (i+ j)

1 1 I I I 1

Figure 5. (3 + 2) + (2 + 3).

(**3)+(*+«)

» — — —

Figure 6. (4 + 3) + (3 + 4) + . . .

r\r\cs\{> Original from

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SYMMETRY

225

(i+i)+0+x)

Figure 7. (2 + 1) + (1 + 2).

Figure 8. (3 + 1) + (1 + 3).

4

Figure 9. 4(3 + 2 + 1) 4(3 + 1 + 2) 4(1 + 3 + 2)

rv i*i fnr»nlf> Original from

Digitizer by VjUU^I^ UNIVERSITY OF MICHIGAN

THEORY OF REGULARITY AND COORDINATION

4U+1+0 4(i+im) 40+i+»)

r

—

» — i

□ □

□ c

i

Figure 10. 4(2 + 3 + 1) 4(2 + 1 + 3) 4(1 + 2 + 3)

4(i+i+0 40+1+0 40*1+1)

a • a a

Figure 11. 4(2 + 1 + 1) 4(1 + 2 + 1) 4(1 + 1 + 2)

4{»+m)
*■

4

H

i —

i — i

Figure 12. 4(3 + 3 + 2) 4(3 + 2 + 3) 4(2 +3+3)

4 (»+!«)

4 f 1+5+2)

4^+i+i)

Digi

Figure iJ. 4(3 + 2 + 2) 4(2 + 3 + 2) 4(2 + 2 + 3)

Original from

ogie

UNIVERSITY OF MICHIGAN

SYMMETRY

227

4(«+m) 4f

4

Figure 14. 4(4 + 3 + 3) 4(3 +4+3) 4(3 +3+4)

4(2 + 5 + 2)

4(1+1+5]

Figure 75. 4(5 + 2 + 2) 4(2 + 5 + 2) 4(2 + 2 + 5)

12
12

— h

4(^+5+ s)

Figure 16. 4(5 + 5 + 2) 4(5 + 2 + 5) 4(2 + 5 + 5)

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

228

THEORY OF REGULARITY AND COORDINATION

4(i+»+5)

4(5 U*3)

Figure 17, 4(2 + 3 + 5) 4(2 + 5 + 3) 4(5 + 2 + 3)

I I

4(5+14-2.)

Figure IS. 4(3 + 2 + 5) 4(3 + 5 + 2) 4(5 + 3 + 2)

.□
"□

4O+3+0

n

□

n

□

40+i+J)

an

□

□

Digi

Figure 19. 5(3 + 1 + 1) 4(1 + 3 + 1) 4(1 + 1 + 3)

Original from
UNIVERSITY OF MICHIGAN

Google

SYMMETRY

229

l+i)

XI

□

r

i

i

4(1+1+4)

Q

n

Figure 20. 4(4 + 1 + 1) 4(1 + 4 + 1) 4(1 + 1 + 4)

1

4-

1+2+3
within

2+3+5
within

3+5+8

Figure 21. Common origin.

t 1

5

L —

Figure 22. Origins uncommon ; superim position of the three trajectories is centered.

Digitized by L:.OOgle UNIVERSITY OF MICHIGAN

230 THEORY OF REGULARITY AND COORDINATION

— i

—

i

Figure 23. 1+2+3+4.

Figure 24. 1+2+4 + 8

4+8

Digitized byGoOgle

Figure 25. 1+2+4 + 8.

Original from
UNIVERSITY OF MICHIGAN

SYMMETRY

h

:

■4*

5 +

ft

—

— .

i

— i —

—

■

j i ' | ■

i

tH— " — i —

j :

i —

i

4(i+3 + 5)

Figure 27. 2 + 3 + 5

2ff. 2 + 3 + 5 + 8

4 +

■

Digiti

Figure 28. 3 4- 5 + 8

Original from
UNIVERSITY OF MICHIGAN

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CHAPTER 10
QUADRANT ROTATION 1

\ /P USIC i n any equal temperament, when it is re corded gra phically in rec-
tangular projection, e xpresses L th^cguivalentjof, musical- gg^fl/jj?n jj } equ al

temperament. Such a geom etrical jjrojection of music is^expressed on a plane,

and as such is subj ej^t_to_quadrant rotation of the plan e through three dimen-
sional space. Rotatio n may b e either clockwis e or c ounterclockwise ^

The conception of time, which is based on the common denominator and
not on the logarithmic series, implies two possible positions: (1) the original,
under zero degrees to the field of vision (parallel to the eyes) ; (2) the 180° position
derived from the first one through rotation around the ordinate axis. Such an
ordinate axis is either the starting or the ending limit of the vertical cross-section
of the graph (duration limits). If the original (zero degree position) is conceived
as a forward motion of music in time continuity, then the respective variation
of it (180° position) is the backward motion of the original, when the ordinate
is the ending limit in time.

The logarithmic contraction of time corresponds to the logarithmic con-
traction of space on the graph — and if our music were not bound to a common
denominator system of measurement, it would be possible to apply such pro-
jection practically. This same form of variation has been known in visual art
since about 1533 A.D., in skillful paintings made by German and Italian artists.
They are based on the principle of angle-perspective and have to be looked at
(that is, held at an angle) from right to left, instead of under the zero angle
to the field of vision.

'From The Schillinger System of Musical Com- lowing: geometric inversion of music consists of
position. Copyright, 1941, by Carl Fischer, Inc. "a" of the original form of the music, to start with;
Reprinted by permission. This is an abbreviated then, as the *'b" inversion, the same thing back-
version of Book III, Chapter 1. wards; as the "c" inversion, the original but back-
wards and upside down; and the "d" version, for-

2 It may be helpful to add at this point the fol- wards and upside down. (Ed J — , "

Digitized ty Google » 2 ri^fig&H

Figure 2, Unknown Master, I0ih Century: St. Anthony of Padua,

Comrttty Mmmm of Modern An, Collection, Jacques Lipchitr. Ptrii

Digiti

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UNIVERSITY OF MICHIGAN

234 THEORY OF REGULARITY AND COORDINATION

By revolving the second position of a musical graph through the abscissa
(which becomes the axis of rotation) 180° in a clockwise direction, we obtain
the third position of the original. The axis of rotation must represent a pt (pitch-
time) maximum and the direction of the third position is backwards upside-
down of the original, and forward upside-down of the second position. Further
180° clockwise rotation of the third position about its ordinate produces the
fourth position, which is the backwards of the third position, the backwards
upside-down of the second position and the forward upside-down of the orig-
inal. The respective four positions will be expressed in the following exposition
through (a), ©, © and (3).

QUADRANT ROTATION

235

236 THEORY OF REGULARITY AND COORDINATION

These four geometrical inversions may be used individually as variations
of a given melody. They may also be developed into a continuity in which the
different positions are given different coefficients. Under such conditions the
recurrence of the different positions is subject to rhythm.

This method of geometrical inversion, when applied to the composition of
melodic continuity, offers much greater versatility — yet preserves the unity more
— than any composer in the past was able to achieve. For example, by comparing
the music of J. S. Bach with the following illustrations, the full range of what
he could have done by using the method of geometrical inversions becomes clear.

In Invention No. 8, from his Two-Part Inventions, during the first 8 bars
of the leading voice (upper part after the theme ends), the first 2 bars fall into
the triple repetition of an insignificant melodic pattern lasting one and one-half
times longer than the entire theme.

it i > L j i j irrrr qif L ^i

Figure 5. J. S. Bach, Two-Part Inventions, No. 8.

Using the method of geometrical inversion (even with a compromise of the
recurrence of the original position), we obtain the following version of thematic
continuity.

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UNIVERSITY OF MICHIGAN

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QUADRANT ROTATION

237

Si

®

©

Inversion of J. S. Bach, Two-Part Inventions, No. 8

238 THEORY OF REGULARITY AND COORDINATION

In some cases geometrical inversions of music give new and often more
interesting character to the original. When a composer feels dissatisfied with
his theme, he may try out some of the inversions — and he may possibly find them
more suitable for his purpose, discarding the original. Such was the case when
George Gershwin* wrote a theme for his opera Porgy and Bess, where position ©
was used instead of the original which was not as expressive and lacked the
character of the later version.

An analysis of well-known works of the composers of the past often throws
new light upon them, revealing hidden characteristics that become more ap-
parent in the geometrical inversions. For example, the harmonic minor scale
combined with certain rhythmic forms produces an effect of Hungarian dance
music. In L. van Beethoven's Piano Sonata No. 8, the first theme of the finale
in its position © reveals a decidedly Hungarian character which is not as notice-
able in its original form. This analysis also discloses that position (3) of the same
theme has a more archaic character than the original, linking Beethoven's music
with that of Joseph Haydn.

i 1 ! i 1 i h uu iVr;r m

j% rr rrr i r r r I Vrr^ "to N
j r r r'-rrrVrrrV Pl " |7J "L

Figure 7. Geometric inversions of L. van Beethoven, Piano Sonata, No. 8, Finale.

{continued)

*In the Musical Courier of Nov. 1 . 1940 Leonard
Liebling, editor, wrote: "After George Gershwin
had written over 700 songs, he felt at the end of
his inventive resources and went to Schillinger for
advice and study. He must have valued both.

for he remained a pupil of the theorist for four
and a half years." Porgy and Bess, which took
Gershwin more than two years of work under his
teacher's supervision, was composed according to
the Schillinger System. (Ed.)

igitized by LiOOglC UNIVERSITY OF MICHIGAN

QUADRANT ROTATION

239

©

®

j Try I J» J* J| l r 1 'T l r rh- J

Figure 7. Geometric inversions of L. van Beethoven, Piano Sonata, No. 8, Finale.

{concluded)

It is possible to plan in advance the composition of melodic continuity
through combining geometrical inversions of the original material with a rhythmic
group pre-selected for the coefficients of recurrence of the different positions.

Rhythm of Coefficients: r^i

Geometrical Positions: (a), <2), (g)

Continuity: 3 ®+ @)+2(E) + 2® + (3)+3(£)

The actual technique of transcribing music from one position to another
may be worked out in three different ways. The student may take his choice.

1. Direct transcription of the inverted positions from the graph into musical
notation.

2. Direct transcription from a complete manifold of chromatic tables rep-
resenting (a) and ((J) positions for all the 12 axes.

rv m C^nr\cs\£> Original from

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240

THEORY OF REGULARITY AND COORDINATION

• * • t
■ J • a

* — *

» r

■ U ;

SB

» r

» *

t=s=

» *■

:

Ml

v*-r : — " 1

(S

Figure 8. Manifold of chromatic tables for (a) and (3).

3. Step by step (melodic) transcription from the original.

The unconscious urge toward geometrical inversions was actually realized
in music of the past through those backward and contrary motions of the original
pattern which may be found in abundance in the works of the contrapuntalists
of the 16th, 17th and 18th centuries. As they did not do it geometrically but
tonally, they often misinterpreted the tonal structure of a theme appearing in
an upside-down position. They tried to preserve the tonal unity instead of
preserving the original pattern. Besides these thematic inversions of melodies,
evidence of the tendency toward unconscious geometrical inversions may be
observed in the juxtaposition of major and minor as the psychological poles. In
reality, the commonly used harmonic minor is simply an erroneous geometrical
inversion of the natural major scale. The correct position (3) of the natural
major scale is the Phrygian scale and not the harmonic minor. The difference
appears in the 2d and 7th degrees of that scale.

In the following examples, d© indicates the upward reading of the (3) scale.

i
t

©

®

d© = ©

5E£

I

Digiti

Figure 9. Inversion of natural major

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UNIVERSITY OF MICHIGAN

QUADRANT ROTATION 241

The effect of psychological contrasts, to which I have referred with regard
to scales, takes place with chord structures and their progressions as well. The
most obvious illustration is a major triad (c — e — g; 4 + 3) with its reciprocal
structure minor triad (c — eb — g; 3 +4). When such a chord is to be inverted
from c as an axis, all pitch-units take corresponding places in the opposite direc-
tion, i.e., c remains constant (the invariant of inversion), e becomes ab, and g
becomes f. Here is a comparative chart of positions (a) and (@) of the chords
commonly known as triads [S(5)] and 7th chords [S(7)].

©

j n || mi H «n y i>n „ ft „ f ft „ fft ,

ii w p ii ■ u i n'iu ipu ii i m iiiiw ii

Figure 10. (a) and (3) positions of the triads.

This method of inverting chords as well as scales in order to find the psy-
chological reciprocal is particularly useful in cases where there is doubt as to what
the reciprocal chord structure or progression may be. It also provides an exact
way of finding the reciprocal structures and progressions in those cases in which
the latter are entirely unknown — and the trial and error method does not bring
any satisfactory result.

The technique of transcribing any harmonic continuity into different geo-
metrical positions can be greatly simplified by using the method of enumeration
of each voice of the harmony. Each voice becomes a melody and it is only neces-
sary to know the entire chord (i.e., the starting-points of such melodies) for the
starting-point, after which all voices may be transcribed horizontally (as mel-
odies).

rv - ■ wh r\r\Cf\f> Original from

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242

THEORY OF REGULARITY AND COORDINATION

®

Jr l.» tro -- k Q '» " i \*** o o n

■

rm-j

Figure 11. (a) and (2) of melody with chords.

The above-mentioned operations make it clear that any of the variations
in the original distribution of voices of a chord may serve as a starting-point for
any harmonic continuity. Thus, a 4-part harmony offers 24 versions in each
of the four geometrical positions. This device is superior to the ingenuity of
any composer using an intuitive method in order to achieve variety of instru-
mental forms of the same harmonic continuity.

The following chart represents 24 original forms of distribution of the
starting chord (according to the 24 permutations of 4 elements), for the har-
monic continuity offered in the preceding figure 11. When the starting chord
has the same structure but different distribution, the resulting sonority of each
version also becomes different.

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QUADRANT ROTATION

243

d c c c
cd bb
bbda
a a a d

dbbb
b d c c
c c d a
a a a d

dbbb
b d a a
a a d c
c c c d

d c c c
c d a a
a a d b
bbbd

d a a a
a d c c
c c d b
bbbd

d a a a
adbb
bbdc
c c c d

43 3 3
3 4 2 2
2 2 4 1
1114

4 2 2 2

2 43 3

3 3 4 1
1114

4 2 2 2

2 4 11
1 1'4 3

3 3 3 4

43 3 3
3 4 11
1142
2 2 2 4

4 111
143 3
3 342
2 2 2 4

4 111
142 2
2 2 43
333 4

Figure 12. Twenty-four original forms of distribution of starting chord.

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CHAPTER 11

COORDINATE EXPANSION 1

"LJAVING DISCUSSED the technique of geometrical inversions, we may now
** consider an additional set of techniques, those leading to geometrical
expansions.

On an ordinary graph, the unit of measurement is equivalent to of an
inch, and it represents, in this system of notation, the standard pitch-unit, i.e.,
\/2 (a semitone). Such units are expressible in arithmetical integers as loga-
rithms to the base of \/2. Thus, a semitone consists of one unit, a whole tone
of two units, etc., along the ordinate.

A melodic graph may be translated into different absolute pitch values by
substituting different coefficients for the original p.

To translate a musical graph into y/2 we would simply use double units
on the ordinate for the original single units, while preserving all the other rela-
tions within" a given melodic continuity. In this case, p = 2p. By using greater
coefficients such as 3, 4, 5, 6 or 7 (\^2, \/2, ty2 h , y/2, \VV), we obtain the
respective units for the pitch intervals.

This form of projection is known as an optical projection through extension
of the ordinate. It is one of the natural tendencies in visual arts. When artists
attempt to produce a distortion (variation) of the original proportions, they are
unconsciously attempting to achieve one or another form of geometrical pro-
jection.

These variations, when executed geometrically and in accordance with
optics, give a greater amount of esthetic satisfaction because they are more
natural.

On the next page you will find an example of the translation of one system
of proportions into another, as applied to linear design.

'Reprinted with permission, from The Schillinger
System of Musical Composition, Copyright, 1941,
by Carl Fischer, Inc. This is an abbreviated form
of Chapter 2 of Book III. (Ed.)

244 Original from

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COORDINATE EXPANSION

245

Figure I. Translating one system of proportions into another.

In the illustration above, the same configuration is presented under different
coordinate ratios. The technique of such translation consists of producing a
network on the original drawing (with as many units as is desirable with regard
to precision) and then transcribing this network into a differently proportioned
area, preserving the same number of lines on both coordinates of the network.
Then all points of the drawing acquire their respective positions in the corres-
ponding places of the network.

Compare these geometrical projections with the distortions in these and

other paintings by El Greco and Modigliani.

n . ... nA ^C* c\C\a\{> Original from

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I

246 THEORY OF REGULARITY AND COORDINATION

Figure 2. El Greco*

1

V

Figure 3. Modigliani**

* Metropolitan Museum of Art, New Totk.
** Collection, The Museum of Modern Art, New York.

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COORDINATE EXPANSION

247

As each coefficient of expansion is applied to music, the original is translated
into a different style, a style often separated by centuries. It is sufficient to
translate music written in the 18th century by the coefficient 2 in order to ob-
tain music of greater consistency than an original of the early 20th century
style. For example, a higher quality Debussy-like music may be derived by
translation of Bach or Handel into the coefficient 2.

The coefficient 3 is characteristic of any music based on %/2 (i.e., the "di-
minished 7th" chord). Any high-quality piece of music of the past exhibits,
under such projection, a greater versatility than any of the known samples that
would stylistically correspond to it in the past. For the sake of comparing the
intuitive patterns with the corresponding forms of geometrical projection, it is
advisable to analyze such works as J. S. Bach's Chromatic Fantasy and Fugue,
Liszt's B Minor Sonata, L. van Beethoven's Moonlight Sonata, first movement.

The coefficient 4, being a multiple of 2, gives too many recurrences of the
same pitch-units since it is actually confined to but 3 units in an octave. Natur-
ally, such music is thereby deprived of flexibility.

But the 5p expansion is characteristic of the modern school which utilizes
the interval of the 4th — such as Hindemith, Berg, Krenek, etc. Music corres-
ponding to further expansions, such as 7p, has some resemblance to the music
written by Anton von Webern. Drawing comparisons between the music of
Chopin and Hindemith, under the same coefficient of expansion, i.e., either by
expanding Chopin into the coefficient 5, or by contracting Hindemith into the
coefficient 1, we find that the versatility of Chopin is much greater than that of
Hindemith. Such a comparison may be made between any waltz of Chopin
and the waltz written by Hindemith from his piano suite, 1922.

Comparative study of music under various coefficients of expansion reveals
that often we are more impressed by the raw material of intonation than by the
actual quality of the composition.

The opposite of this procedure of expansion of pitch is contraction of pitch.
Any pitch interval-unit may be contracted twice, three times, etc., which is
expressible in *$/2, ^2, etc., providing that instruments with corresponding
tuning are devised. Those esthetes who usually love to talk about the "economy
of material" and "maximum of expression" will perhaps be delighted to learn
that an entire 4-part fugue of Bach occupying a range of 3-£ octaves would re-
quire only one whole tone if the pitch interval-unit were -j^ of a tone C^yi).
Applying the same principle to the contraction of the absolute time duration-
unit, we could hear this fugue in a few seconds instead of several minutes!

The natural pitch-scale, i.e., the series of harmonics, does not produce uni-
form ratios but gives a natural logarithmic contraction. The intervals between
the pitch-units decrease, while the absolute frequencies increase. This phe-
nomenon is analogous to the perspective contraction in space as we see it. If
music were devised on natural harmonic series, the relative group-coefficients of
expansion and contraction could be used. But it seems that the natural harmonic
series does not, in fact, provide any flexible material for musical intonation but
merely for building up various tone qualities — for the fact is that a group of
harmonics sounded at the same intensity produces one saturated unison rather

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248 THEORY OF REGULARITY AND COORDINATION

than harmony. This phenomenon is somewhat similar to that of white light,
in which all sprectral hues merge — becoming noticeable only when the beam is
broken up. Logarithmic contraction of pitch combined with the logarithmic
contraction of time may come into existence in the remote future in connection
with the development of automatic instruments for composition and execution
of music.

The technique of pitch -expansion may be executed directly from a graph
or from a corresponding chromatic scale of expansion. In such a case, 2p will
produce a whole tone scale progressing through 2 octaves instead of a full chro-
matic scale progressing through one octave (when p = 1.) While expansion of
time extends the graph along the abscissa, the increase of the absolute time unit
is not noticeable unless compared with the original. When we hear a musical
continuity, we do not know (unless it is extremely exaggerated) whether it is
the original velocity or a derivative thereof. The difference becomes apparent
only when different coefficients of velocity of the same musical continuity are
brought close together. Thus, time extension produces a different pattern on
a graph without producing a difference detectable in the absence of comparison.

Pitch expansion works under the same conditions. It is only through com-
parison that we can learn that a certain musical continuity has been expanded
or contracted from its original. This is apparent in the process of tonal expansion^
(which preserves all the pitch-units while the range increases).

JiilinilJ

._ „ .__ : * — *

jj.

: p2t ::::

— itiij T; " --

::::::::::::::::

; :r — zz \

tTTrn i ■ H4 f H4ti i -fflfflRffl+H

x ,

---■1

p^t

[I : , 1 _______ - .-"I.

Figure -4. Time and pitch expansions (continued).

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COORDINATE EXPANSION

249

— Mill

t2p

t3p

t

a

2t2p

E

3t3p

Figure 4. Time and pitch expansions (concluded)

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250 THEORY OF REGULARITY AND COORDINATION

If pt represents the original, 2t and 3t produce the corresponding time ex-
pansions. Likewise, 2p and 3p produce the corresponding pitch expansions.
The expansion through two coordinates preserves the absolute form of the
configuration, merely magnifying it (2p2t and 3t3p).

It might seem at first that the ordinary enlargement or reduction of an
original image — such as that effected by any natural optical projection (lantern
slide projector, motion picture projector, magnifying glass, etc.) — does not
change the appearance of the image. Yet when carried to an extreme, h does
in fact transform the image to a great extent. For example, an ordinary close-up
of a human head seen on the screen does not change our impression of the image.
But when a human head is subjected to a several hundred power magnification,
the original image is changed beyond recognition. A photograph of the skin
surface of the human arm occupying only l/100th of a square inch produces an
image which is not easily associated with the human arm.

Thus, the difference in the actual sound of music (like the magnification of
Haydn into von Webern) is only quantitatively different from the enlarging of
visual images. Even with coefficients as low as 5, a melody is transformed beyond
recognition. But the magnification of visual images requires at least one-hundred
power magnification in order to achieve a similar effect.

It is interesting to note that bizarre effects of optical magnification are
often due to the fact that such images are merely hypothetic and have no actual
correspondences in the physical world of our planet. An image of a chicken can
be magnified to the size of the Empire State building (for example, by being
projected on an outdoor smoke screen), yet no real live chicken could exist on
this planet even the size of an ostrich, because — as the volume grows in cubes —
the legs of such a chicken could not support the weight of its body.

The following chart represents pitch expansions of the melody: graphed
in Figure 4.

Figure 5. Pitch expansions of the melody of Figure 4 (continued).

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COORDINATE EXPANSION

251

4p

[im, P 1

^ 1_

1 i h r 1

fir — i

p 1

^•"j r 1

Figure 5. Pitch expansions of the melody of Figure 4
(concluded).

Geometrical expansions oi melody may also serve the purpose of modifying
motifs through the method of geometrical projection. The original melodic
pattern becomes entirely modified — yet the system of pitch-units is the outcome
of a consistent translation from one system of pitch relations to another. The
technique of such modification is equivalent to the contraction of the general
pitch range emphasized by the geometrically expanded form. Some melodies,
especially those with big coefficients of expansion, permit several different
versions (degrees) of contraction.

The following example presents the exact geometrical expansions with the
respective contractions of their ranges:

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252 THEORY OF REGULARITY AND COORDINATION

7p

1

Readjusted range

Figure 6. Geometrical expansions with readjusted {contracted) range.

The process of range-contraction often introduces new characteristics into
geometrically expanded forms. For example, in the case of 5p in the preceding
example: in its readjusted form, it seems to be more "conservative" than in
its respective geometrical expansion. In the case of 7p, the contracted form is
reminiscent of the music of Prokofief rather than that of von Webern.

Geometrical expansion of the harmony which accompanies melody expanded
through the same coefficient (whether with readjusted range or not), must be
performed from the pitch axis of the entire system (usually the root-tone).

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COORDINATE EXPANSION

253

Figure 7. Geometrical expansion of a harmonized melody.

This translation of harmony may be accomplished either through transcription
of a graph or through step by step translation from the original. One may also
prepare in advance chromatic scales from the respective pitch axes where all
the pitch-units may be found directly in the corresponding expansions.

Figure 8. Scale of pilch- units and their corresponding expansions.

All geometrical expansions are subject to geometrical inversions as well.
A consistent musical continuity may be evolved through the variation of in-
versions under the same coefficient of expansion. Thus the two methods of
mathematical variation of music, based on geometrical projection, bring an
effective solution to two very important technical problems:

1. Composition of infinite melodic or harmonic continuity containing or-
ganically related contrasts.

2. Translation of music of one epoch into another, "modernization" and
"antiquation."

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UNIVERSITY OF MICHIGAN

CHAPTER 12
COMPOSITION OF DENSITY 1

The behavior of sounding texture in any musical composition is such that
it fluctuates between stability and instability, and so remains perpetually in a
state of unstable equilibrium. The latter is characteristic of albumen which
is chemically basic to all organic forms of nature. For this reason, unstable
equilibrium is a manifestation of life itself, and, being applied to the field of
musical composition as a formal principle, contributes the quality of life to music.

NOMENCLATURE:

d — density unit = p, S

D — simultaneous density-group = S, 2S, ... 2.
D — sequent density-group (consecutive D)

A (delta) — compound density-group representing density limit in a given score

(simultaneous A = 2)
A (delta) — sequent compound density group: general symbol for the entire

consecutive composition of density: A - * = 2
A~~* (A - *) — *■ the delta of a delta: sequent compound delta.
^ (phi) — individual rotation-phase:

<t> O and $ O in reference to t or T
<t> C) and ^ Q in reference to p or P, or d or D
9 (theta) — compound rotation-phase, general symbol of the continuity of rotary
groups in a given score; it includes both forms of 4>.

'From The Schillinger System of Musical Com-
position, Copyright, 1941, by Carl Fischer, Inc.,
by whose permission it is reprinted. This is an
abbreviated version of Chapter ISof BooklX. (Ed.)

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COMPOSITION OF DENSITY

255

A. Technical Premise

Depending on the degree of refinement with which the composition of density
is to be reflected in a score, d may equal p or S. In scores predominantly using
individual parts, either as melodic or harmonic parts, it is possible and advisable
to make d = p. In scores of predominantly contrapuntal type, where each
melody is obtained from a complete S, d = S is a more practical form of assign-
ment.

One of the fundamental forms of variation of the density-groups is rotation

The abscissa (horizontal) rotation follows the sequence of harmony (CorC);
in it, all pitch-units (neutral or directional) follow the progression originally
pre-set by . harmony.

The ordinate (vertical) rotation does not refer to vertical displacement of
p or S, but to thematic textures (melody, counterpoint, harmonic accompani-
ment) only; therefore there is no vertical rearrangement of harmonic parts at any
time. Such displacement of simultaneously correlated S would completely change
the harmonic meaning and the sounding characteristics of the original. Tech-
nically such schemes arc possible only under the following conditions:

(1) identical interval of symmetry between all strata;

(2) identical structures with identical number of parts in all strata.

The above requirements impose limitations which are unnecessary in or-
chestral writing, as it means that each orchestral group would have to be re-
presented by the same number of instruments, which is seldom practical.

The idea of bi-coordinate rotation (i.e., through the abscissa and through
the ordinate) implies that the whole scheme of density in a composition first
appears as a graph on a plane, then is folded into a cylindrical (tubular) shape
in such a fashion that the starting and the ending duration-units meet, i.e.,
A~~ * = limiti «-» t m . Under such conditions the cylinder is the result of bending
the graph through ordinate, and the cylinder itself appears in a vertical position.
Variations are obtained by rotating this cylinder through abscissa, which cor-
responds to C and 4> C.

Therefore: A - * = C (t, — tj, </> G (t m — t,).

Folding the scheme of density (as it appears on the graph) in such a fashion
that the lowest and the highest parts of the score meet, we obtain the limits
for p, i.e., A = lim pi *-* p m . Under such conditions the cylinder is the result
of bending the graph through abscissa, and the cylinder itself appears in horizontal
position. Variations are obtained by rotating this cylinder through ordinate,

which corresponds to $0 and ^() Therefore : A - * » <t>0 [ T ). 4> O I 1 ) •

Here delta is consecutive as physical time exists during the period of rotation.

of phases.

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256

THEORY OF REGULARITY AND COORDINATION

B. Composition of Dknsity-Gkoui-s

As we have mentioned before, the choice of p and t, or of S and T as density
units, depends on the degree of refinement which is to be attributed to a certain
particular score. For the sake of convenience and economy of space, we shall
express dt as one square unit of cross-section paper. In each particular case, d
may equal p or S, and T may equal t or mt. Yet we shall retain the dt unit of
the graph in its general form.

Under such conditions a scale of density-time relations can be expressed
as follows:

D

= d, D = 2d, .

. . D

= m:l

D~>

= dt, D~~* = d2t, .

. . D - *

= dmt

D~>

= dt, D~* = 2dt, .

. .

= mdt

D-

= dt, D~* = 2d2t, .

. . D~*

= mdnt

The above are monomial density-groups. On the graph they appear as follows:

m

T

D~* = dt

D = d2t

= d3t

etc.

D~

= 2dt

P~* =

= 3dt

I

)^ =

= 4dt

et

c.

D~* = 2d

2t

D~* = 2d

3t

D~* = 3d2t

etc.

1

Figure 1. Monomial density-groups.

Binomial density-groups can be evolved in a similar way:

A~~ * = D7~* + DJ~*; Dr=dt: = 2d2t;

A - * = dt + 2d2t

Figure 2. Binomial density-groups {continued.)

■ | f~\r~\ cs I c> Q i n a I rro m

ngitized by ViWgi v UNIVERSITY OF MICHIGAN

COMPOSITION OF DENSITY 257

= 07* + Dp; Dp - d2t; Dr* = 2dt;

A - * « d2t + 2dt

£~* = Dp + Dp; Dp - 2d3t; Dp = 5d3t;

A~* « 2d3t + 5d3t

Figure 2. Binomial density-groups {concluded).

Polynomial density-groups may be evolved, depending on the purpose,
from rhythmic resultants, permutation -groups, in volution -groups, series of vari-
able velocities, etc.

A - " - Dp + Dp + Dp; Dp = 3d3t; Dp = dt; Dp = 2d2t;

ST* = 3d3t + dt + 2d2t

A~~ ^ = 4D^ ; Dp = 2d4t + 2d2t + 2d2t + 4d2t

Figure 3, Polynomial density-groups.

r\r\cs\(> Original from

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258 THEORY OF REGULARITY AND COORDINATION

As it follows from the above arrangement of density-groups, the latter may
be distributed in any desirable fashion, preferably in a symmetric one within
the range of D.

A~* = 5D~>; D7* = d8t; Dp = 2d5t; = 3d3t; D7* = 5d2t; = 8dt;
A - * = d8t + 2d5t + 3d3t + 5d2t + 8dt

Another variant of the same scheme:

Another variant of the same scheme:

Figure 4. Variants of A - * = 5D~*

In all the above cases A > D,i.e., the compound density-group is not greater
than any of the component density groups.

rv -■ _j l r\rscs\f> Original from

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COMPOSITION OF DENSITY

259

Density groups may be considerably smaller than A, in which case there
are many more possibilities for the distribution of D's.

A = 6D; A~* = 4D; Dp = 2d2t; DJ* = dt; DJ* - dt; = 2d2t;
A~* = 2d2t + dt + dt + 2d2t

i —

A

01

o

r: -

■ el

C ■

Figure 5. Density groups smaller than A,

The different distributions as in the above Figure can be specified by means
of their phasic positions.

If we assume that the lowest d of A designates fa i.e., the zero phase, then
0i, 0], . . . designate all the consecutive phases. Thus the first variant of
Figure 5 can be expressed as follows:

A - * = (2d2t)0o + (dt)0, + (dt}0, + (2d 2 1) 04.

— 1

— p-

-0J

H

V 3

1— «1 J

— 0.H

ir

Figure 6. First variant of Figure 5.

It follows from the above that the first (&0 and the last (4>«) phases are
identical.

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UNIVERSITY OF MICHIGAN

260 THEORY OF REGULARITY AND COORDINATION

C. Permutation of Sequent Density-Groups within the Compound
Sequent Density-Group

(Permutations of D~* within A~~ *)

Continuity where permutations of D — *'s take place can be designated as a
compound sequent group consisting of several other compound sequent density
groups, the latter being permutations of the original compound group. Then
such a compound density-group yielding n permutations of the original compound
sequent density group can be expressed as follows: A - * (A - *) = AT* + A| +

+ AT + ... AT.

AT* - (3d3t) DTVo + (dt) DT^o + (2d2t) DT**> + (2d2t) DT**i +
+ (dt) D7"\fr, + (3d3t) DT**o, where A = 3d.

A~* (A - *) C = AT* + A"T* + AT* + AT* + AT* + AT* =

- (DT* + DT* + Dp + DT* + DT* + DT*) +
+ (DT* + DT* + DT* + DT* + DT* + DT*) +
+ (DT* + DT* + DT* + DT* + DT* + DT*) +
+ (DT* + DT* + DT* 4- DT* + DT* + DT*) +
+ (DT* + DT* + DT* + Dp + DT* + DT*) +
+ (DT* + Dp + DT* + DT* + DT* + DT*)
See Figure 7 on opposite page.

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COMPOSITION OF DENSITY

261

T

T

3 "

1 ■

q~

q

•

-1

-1

r

s

64

K

t3

a
s

■ex.

3

The same technique is applicable to all cases where A > D, i.e., where delta
is greater than any of the simultaneous density-groups.

rv • ■ wh nnnlp Original from

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262 THEORY OF REGULARITY AND COORDINATION

D. Phasic Rotation of A and A~~ ' through t and d

Assuming A~* = D = dt, we can subject it to rotation:

(1) A - * e = tfa + t<£, + . . . and

(2) A~*0 = dtfx, + d0, + . . .

The following represents scales of rotation for A~~ * T~ * = D T
A = 4d; T~* = 4t; <p 4 = fa. T~* symbolizes the range of duration of D.

The original position: d^n t^xi

The sequence of rotary phases of d :

A^e = d<fo t<fo + d0i t<j><> + d<*> 2 t<fo + d<M<fr>:

The sequence of rotary phases of t :
A - *0 = d<fo t<fo + d<fr> t#i + d^o t<t> 2 + d<k> t<t> 3 :

The sequence of rotary phases of dt:
A - *9 = d^o t^o + d#i t<^i + d</> 2 t</> 2 + d<t> 3 t<fo:

Figure 8. Phasic rotation of A and A~~ *

n\ n . t ,^^C nna\(> Original from

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COMPOSITION OF DENSITY
The same technique is applicable to a A~~ * of any desirable structure.

For example: A - * = 3D - *; D7* = 3d3t; DT* = dt; DT* = 2d2t;

A = 3d AT* = (3d3t + dt + 2d2t)<fc>;

T * = 6t AT* = AT*0i; AT* = AT*02; AT* = ATVa; • . .

eC and O A„ = (3d + d + 2d)<fc,; -

Ai = A o 0i; A 2 = A o 2 ; Aj = A o 0j; . . .

Let A^A - *) 0= AT* (3d0o3t0o + d<M<fc> + 2d0o2t0o) +
+ AT* (3d0,3t0o + d<M0o + 2d0,2t0o) +
+ AT* (3d0 2 3t0o + d0 2 t0o + 2d<M(fc>) •

Then, A~~ *(A~~ *) 9 = AT* -H AT* + AT* appears as follows:

263

— 1 —

A ►

1

A *

r-1 —

: a-»

->

—J

d0,

d*. Q

Let further A~*(A~"*) 9 = AT* (3d<Mt<fo + dtfot^o + 2d0o2t0o) +

+ AT* (3d0 o 3t</> 1 + d6>t*i + 2d0o2t0,) +
+ AT* (3d</>o3t02 + d0ot0 2 + 2d«/>o2t0 2 ) +
+ AT* (3d0o3t0, + d0ot0, + 2d0o2t0,) +
+ AT* (3d<Mt0« + d<fc,t0 4 + 2d0o2t0 4 ) +
+ AT* (3d* 3t*» + d<fc>t0 4 + 2d0o2t0 s ) .

Then, A~~ *(A~~ *) 9 = AT* + AT* + AT* + AT* + AT* + AT* appears as follows:

4-

4-

AT*

A"

! -

-—I

i -

-

it,.

—

. $

-h

-4-

2 _
1

_ A * -

I

&

-0

.1

I

i —

Figure 9. Phasic rotation of A - * = 3D~ * (continued).

Digitized by LiOGglC UNIVERSITY OF MICHIGAN

264 THEORY OF REGULARITY AND COORDINATION

Now wc shall combine the 0() and the OC.

Let A~*(A"~*) 6= AT*D0 o T~*e ( + AT*D0iT~*0, + AT*De,T~ , 1 +

+ A^DeoT^e, + a^do^ - ^ + at'doiT - *© s

Then A - *(A — *} 9 — AT* + AT* + AT* + AT* + AT* + AT* appears as follows:

Figure 9. Phasic rotation of A - * — 3D~~* (concluded).

The diagonal and vertical lines are inserted for clarity.

The addition of positive or negative phases of rotation to any given position
of A~~ * follows the rules of algebraic addition. Thus if the given position is fa,
the addition of one or C) brings the density-group into position fa, or:
fa 4- 4> = Likewise + 2<f> = fa, fa + m<t> — fa n .

As the last phase equals the first phase, or = fa, negative quantities of
phases, or the counterclockwise phases, i.e., <p'C< or 4>0, must be added with
the sign minus to the last phase. Thus if the given position is fa and the number
of phases is n, the addition of one negative phase brings the density-group into
position n _i; or, n — </> = <t> n i- Likewise, 4> n - 2</> = n _ 5 , m<£ = <f>„_ m .

Problem: find the phase 4> after the following forms of rotation have been
performed from the original fa, where = 8<£: 2<f> — S<f> + 5<p + $ — 4<f> + 3$ —

- *■

Solution: fa « fa + 24, - 3* + 5<f> + 4> - 4> + 3* - * = fa + 11* -

— &4> = fa + 34> = fa, i.e., the density group appears in its third phase.

This is applicable to both ordinate and abscissa. It follows from the above
reasoning that in order to obtain the original position D , after performing a
group of phasic rotations, the sum of the coefficients of 4> must equal zero. As

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COMPOSITION OF DENSITY 265

we know from the theory of rhythm, all resultants with an even number of
terms have identical terms in both halves of the resultants. If such terms, used
as coefficients of 0, are supplied with alternating "plus" and "minus", the sum
of the whole resultant would be zero. This gives a perfect solution for the cases
of variation of density groups, because resultants, being symmetric, produce a
perfect form of continuity.

Examples:

T4-1-3 = 3 + 1+2+2 + 1+3; changing the signs, we obtain:

3- 1+2-2 + 1-3 = 6-6 = 0.

r5-i-4 = 4 + 1+3+2 + 2+3 + 1+4; changing the signs, we obtain:

4- 1+3-2+2-3 + 1-4 = 10-10 = 0.

e(r 7 -i-2) = *> + 2<> - 20 + 20 - + <t> - 20 + 2* - 20 =
= 0n + 70 -70 = 0o+O = 0o.

->

Figure 10. A pplying resultants from the theory of rhythm.

Computation of the phasic position 8 X , which is the outcome of a group of
phasic rotation, can be applied to any position 6 m to which such rotations have
been applied. The computation is performed through the use of same technique
as before, i.e., through algebraic addition.

The technique of phasic rotation of the density-groups can be pursued to
any desirable degree of refinement. The phases of d and t can be synchronized
when they are subjected to independent rotary groups, in which case we follow
the usual formula:

Od = eh\ G't (Od)

Ot ~ O't O'd (Ot)

rv -■ _j l r\rscs\{> Original from

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266 THEORY OF REGULARITY AND COORDINATION

In composing the original density-group (Ao'TjJ - *), it is important to take
into consideration the character of j relations with regard to the effects such
relations produce. In this respect we can rely on the three fundamental forms
of correlation, i.e., the parallel, the oblique and the contrary.

When they are applied to density-groups, these three forms must be inter-
preted in the following way:

(1) parallel: identical ratios of the coefficients of <£d and 0t;

(2) oblique: non-identical ratios of the coefficients of <f>d and </>t, where —

(a) partial coincidence of the coefficients takes place, and/or

(b) the coefficient of one of the components (either d or t) remains
constant ;

(3) contrary: identical ratios arranged in inverted symmetry. When the
number of coefficients in both coefficient-groups is odd, such case should
be classified as oblique, due to partial coincidence of coefficients.

E. Practical Application of £~~ * to 2 — \

(Composition of Variable Density from Strata)

In its complete form, this subject belongs to the field of Textural Composi-
tion and will be treated in this chapter only to the extent necessary in order to
make the whole subject more tangible.

The first consideration is that A~~ * can be composed to a given 2 , or 2
can be compossd to a given A~~\ This means that either a progression of chords
in strata or a density-group may be the origin of a whole composition. One
harmonic progression may be combined with more than one density-group; the
opposite is also true, i.e., more than one harmonic progression can be written
to the same group of density. For this reason the composer's work on such a
scheme may start either with 2~~* or with A - *.

It is practical to consider d — S as the most general form of the density-
unit, leaving d = p for cases of particular refinement with regard to density.
If d = S it means that one density-unit may consistof p, 2p, 3p or 4p. In actual-
ity, however, harmonic strata acquire instrumental forms, in which case even
S4p may sound like rapidly moving melodies. On the other hand, S may be
transformed into melody, in which case we also hear one part.' The implication
is that, in the average case, the density of a melodic line and the density of
harmony subjected to instrumental figuration are about the same. Physically
and physiologically, and therefore psychologically, density is in direct proportion
to mobility. This means, for example, that a rapidly moving instrumental form
of successive single attacks, which derives from S4p, is nearly as dense as a
sustained chord of S4p; the extreme frequency of attacks makes an arpeggio
sound like a chord, i.e., in our perception, lines aggregate into an assemblage.

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UNIVERSITY OF MICHIGAN

COMPOSITION OF DENSITY

267

The technique of superimposltion of A~~ * upon 2~ * consists of establishing
correspondences between 0d and p, and between <t>t and H, i.e., between the
density-phase, or density-unit, and the number of harmonic parts; and between
the duration-phase, or duration-unit, of the sequent density-group and the
number of successive chords.

All subsequent techniques pertain to composition of continuity, i.e., to
coordination of attacks and durations, instrumental forms, etc.

We shall now evolve an illustration of A~~ * correlated with 2 . To demon-
strate this technique beyond doubt, we shall use the most refined form of it,
where d = p and t = H.

If Nt = NH, then the cycle of A~~ * and 2 — * are synchronized a priori;
Nt

otherwise, (i.e., if j^pj ^ 1) they have to be synchronized. This shows that
with just a few chords and a relatively brief scheme of density, one can evolve
a composition of considerable length, since A - ¥ itself, in addition to interference
with H~" * of 2~~ , can be subjected to rotational variations.

Let the original A^ = A = 8d.

Let AT* = A <fc>2t + d<fot + 5d*o2t + 3d^jt + 2d<fo2t.
As A = D = 8d, 2 must equal 8p.

T~~* = 8t and would require H *= 8H, unless we wish to introduce a
case in which £p=^ 1-

We shall introduce such a case.

Let = SH. Then ™ = g
Hence, T - * 1 - 8t-5 - 40t.

As we intend to use 5 variations of A - *, the entire cycle will be synchronized
(completed) in the form: A - *(A~*) = 40t 40H, where H~~ * ( = 5H) appears 8
times.

For the sake of greater pliability of thematic textures, it is desirable to
pre-set a directional sigma.

We shall choose the following sigma: 2 = Si2p + Sn3p + Sm3p and
2~~* - 5H.

Let I = 3i + 2i + 3i + 5i and 1(2) = -

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UNIVERSITY OF MICHIGAN

268 THEORY OF REGULARITY AND COORDINATION

We shall now subject the £^Q,£^* to variations of density

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UNIVERSITY OF MICHIGAN

COMPOSITION OF DENSITY

269

■ L At 0i

H 4

Figure 12. Variation of density of figure 11. A"7*Bi.
L L ^ 2

H& Hj H a H 3 H* Hg

Figure 13 Variation of density of figure 11 . £T{*Q i.

=<J by ^.UUgl^ UNIVERSITY OF MICHIGAN

270 THEORY

OF REGULARITY

AND

COORDINATION

$

tab

1Mb

tab

s.

Hi H a H3 H4 H5 Hi H2 H3 H4 H5

Figure 14. Variation of density of figure 11. A7*6i.

ft

St ©7

bi

is

tab

i^b

t

8,

pi

pi

Si

1=

Hi H2 Hg H4 H5 Hi Hg Hg Hi H5

rv -■ «wh c\C\C%\c> Original from

Fi t un,5. Variation of ^W^W

PART THREE
TECHNOLOGY OF ART PRODUCTION

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

Consecutive Selective Processes

1 . Construction of a rational set from a real set by selecting the rational numbers
and omitting the real.

2. Selection of a limit between two numbers of a rational set (cycles, wave-

3. Logarithmic selection (exponent scale) of points within the given limits.

4. Establishing symmetric points within the logarithmic scale.

5. Establishing the forms of group-symmetry (binomial, trinomial, polynomial)
within the logarithmic scale.

6. Developing the secondary selection: operand groups =. artistic scales (assy met-
ric and symmetric).

7. Variation of the operand group

(a) phase displacement (circular permutation of elements);

(b) general permutation of elements;

(c) quadrant rotation;

(d) G () and (), C and 9 O a "d their combinations (vari-
ation through coordinate rotation) applied to a compound operand
group.

8. Composition of components of the compound operand groups through the
process of two-coordinate development.

lengths [Angstrom units]).

Original from
UNIVERSITY OF MICHIGAN

CHAPTER 1

SELECTIVE SYSTEMS

A. Primary and Secondary Selective Systems 1

T^HE logarithmic dependence of ratios within a given limit-ratio is one of the

more complex forms of dependence in the field of uniformity. Within the
limit of £ ratio, for example, we can establish a scale of uniform n ratios. These
may be represented on a straight line, the extension of which would correspond
in frequencies to the ratio of by equidistant symmetric points. The first
point of such linear extension would correspond to b, and the last point, to a.
All the intermediate points of uniform symmetry would thereupon be repre-
sented as follows:

Such sets of uniform ratios may be regarded as primary selective systems.
The number of points in a straight line, which is finite in itself, is infinite. The
possible points in a line would thus be represented by irrational as well as rational
number values. Another way of expressing this concept is to say that all of the
possible number values in a line constitute a "dense set." Primary selective
systems, as equidistant symmetric points, are not "dense sets."

Non-uniform forms of regularity, as we have previously shown, are the
resultants of the interference of two or more uniform periodic waves of different
frequencies brought into synchronization. These resultants, the parent shapes
of all rhythms and the source of configurations, may be obtained either by direct
computation or through graphs. When these resultants are applied in direct
sequence to any of the primary selective systems, they, in turn, produce secondary
selective systems.

As in the case of primary systems, secondary selective systems vary in
density. When a secondary system reaches a point of saturation, it becomes
identical with the parent primary system. When the primary system becomes
saturated, it merges with the continuum. Thus, we may state that secondary
selective series are the result of rarefying primary selective series, which con-
stitute the dense set of the secondary.

'See Appendix C, which presents a primary linger called it Double Equal Temperament to

selective system (tuning system) in music worked distinguish it from our equal temperament system,

out by Schillinger for the execution of intonations (Ed.)
not possible in our present tuning system. Schil-

a

b

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UNIVERSITY OF MICHIGAN

274

TECHNOLOGY OF ART PRODUCTION

In linear design, secondary selective systems are the source of sequences of
linear, plane, or solid motion, which result either in static configurations like
spirals or polygons, or in trajectories of various types that include time as a
component. In music, secondary selective systems produce sequences known as
the rhythm of durations, of pitch (i.e., pitch scales), of chord progressions, of
intensities, of qualities, and of attacks.

The refinement of primary selective systems depends on the discriminatory
capacities of perception. As sight is a more developed form of sensation and
orientation, it permits the construction of primary selective systems that are
denser sets than in the case of auditory orientation.

The primary selective system, which dominates the music of the western
world, is known as equal temperament and consists of a series of 12 semitones.
It is possible to construct a tuning system which would permit execution of other
systems of intonation, such as mean temperament, just intonation, and the
inflections of special types of intonation. The author has devised a system of
tuning, "double equal temperament," which successfully unifies these systems
of intonation. (See Appendix C.)

The material presented in the succeeding pages comprises secondary systems
that are of fundamental importance in the various arts. The temporal scales, as
the parent shapes of all rhythms, are useful in all the arts. The pitch scales are
primarily of importance in music. The scales of linear configuration and the
color scales apply to the graphic arts.

We measure time through the use of a clock system, which is based on a
sequence of uniform moments corresponding to the set of natural integers. In
the field of auditory association, as aroused by the ticking of a clock or metro-
i nome, such uniform time intervals, or periodicities, may be regarded as a primary
; selective system. Secondary selective systems may be constructed by abstracting
^durations from the uniform set, or by causing interference between two different
Viniform series. The rhythmic resultants constitute temporal scales, which are
the basic forms of regularity and coordination, i.e., of rhythm, in all the arts.
I The reader is referred to Appendix A, which contains a detailed presentation
qf the forms of regularity and coordination. Among the rhythmic resultants
presented in Appendix A are the following: (1) binary and ternary synchroniza-
tion; (2) distributive involution groups; and (3) groups of variable velocity.

In Appendix C the reader will find a description of the Rhythmicon, the
first modern instrument for composing music, or temporal scales, automatically.
The rhythmic resultants produced by this instrument are based on the inter-
ference of generators from one to sixteen .

. Temporal Scales

Digit

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UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS 275
1. Rhythmic Resultants.

r 3+2 - 2 + 1 + 1 + 2

r 4+3 = 3 + 1 + 2 + 2 + 1 + 3

r a+2 = 2 + 2 + 1 + 1+ 2 + 2

r 8+3 = 3 + 2 + 1+ 3 + 1 + 2 + 3

r 5+4 = 4 + 1+ 3 + 2 + 2 + 3 + 1+ 4

r 6 + 5 = 5 + 1+ 4 + 2+ 3+ 3 + 2 + 4 + 1+ 5

r 7+2 = 2 + 2 + 2 + 1 + 1+ 2 + 2 + 2

r 7+3 = 3 + 3 + 1+ 2 + 3 + 2 + 1+ 3 + 3

r 7+4 = 4 + 3 + 1+ 4 + 2 + 2 + 4 + 1+ 3 + 4

r 7+6 = 5 + 2 + 3 + 4 + 1+ 5 + 1+ 4 + 3 + 2 + 5

r 7+6 = 6 + 1+ 5 + 2 + 4 + 3 + 3 + 4 + 2 + 5 + 1+ 6

r 8+3 = 3 + 3 + 2 + 1+ 3 + 3 + 1+ 2 + 3+ 3

r 8+5 = 5+ 3 + 2 + 5 + 1+ 4 + 4 + 1+ 5 + 2+ 3 + 5

r 8+7 = 7 + 1+ 6 + 2 + 5 + 3 + 4 + 4 + 3 + 5 + 2 + 6 + 1+ 7

r 9+2 = 2 + 2 + 2 + 2 + 1 + 1+ 2 + 2 + 2 + 2

r 9+4 = 4 + 4 + 1+ 3+ 4 + 2 + 2 + 4 + 3 + 1+ 4 + 4

r 9+6 = 5 + 4 + 1+ 5+ 3 + 2 + 5 + 2+ 3 + 5 + 1+ 4 + 5

r 9+7 = 7 + 2 + 5 + 4 + 3 + 6 + 1+ 7 + 1+ 6 + 3 + 4 + 5 + 2 + 7

r 9+8 = 8 + 1+ 7 + 2 + 6 + 3 + 5 + 4 + 4 + 5 + 3 + 6 + 2 + 7 + 1 + 8

2 . Rhythmic Resultants with Fractioning.
r 3*2 = 2 + 1+1 + 1 + 1 + 1+ 2

1-4*3 = 3 + 1+ 2 + 1 + 1 + 1+1+2 + 1+ 3

r-6+2 - 2 + 2 + 8(1) + 1 + 8(1) +2 + 2

C f\r\ci\{> Original from

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276 TECHNOLOGY OF ART PRODUCTION

iV^ = 3 + 2 + 1 + 2 + 4(1) + 1 + 4(1) + 2 + 1 + 2 + 2

1-5+4 = 4 + 1+ 3 + 1 + 1+ 2 + 1+ 2 + 1 + 1+ 3 + 1+ 4

le+s = 5 + 1+ 4 + 1 + 1+ 3 + 1+ 2 + 2 + 1+ 3 + 1+1+4 + 1 + 5

= 2 + 2 + 2 + 18(1) + 1 + 18(1) + 2 + 2 + 2

r 7+3 = 3 + 3 + 1+ 2 + 1+ 2 + 12(1) + 1 + 12(1) + 2 + 1 + 2 + 1 +
+ 3 + 3

r LH = 4 + 3 + 1+ 3 + 1+ 2 + 1 + 1+ 2 + 6(1) +1 + 6(1) + 2 +
+1+1+2+1+3+1+3+4

1-7+5 = 5 + 2 + 3 + 2 + 2 + 1+ 2 + 2 + 1 + 1+1+2 + 1+ 2 + 1 +
+1+1+2+2+1+2+2+3+2+5

ri±i = 6 + 1+ 5 + 1 + 1+ 4 + 1+ 2+ 3 + 1+ 3 + 2 + 1+ 4 + 1 +
+1+5+1+6

1-8+3 = 3 + 3 + 2 + 1 + 2 + 1+ 2 + 18(1) + 18(1) + 2 + 1 + 2 + 1 +
+2+3+3

rs+8 = 5 + 3 + 2 + 3 + 2 + 1+ 2 + 2 + 1+ 2 + 1 + 1 + 1 + 1+ 2 +
+1+1+1+1+1+1+1+1+2+1+1+1+2+1+
+2+2+1+2+3+2+3+5

rg+7 = 7 + 1+ 6 + 1 + 1+ 5 + 1+ 2 + 4 + 1+ 3 + 3 + 1+ 4 + 2 +
+1+5+1+1+6+1+7

ro+2 = 2 + 2 + 2 + 2 + 32(1) + 1 + 32(1) + 2 + 2 + 2 + 2

r£±1 = 4 + 4 + 1+ 3 + 1+ 3 + 1+1+2 + 1 + 1+ 2 + 16(1) + 1 +
+ 16(1) + 2 + 1 + 1+ 2 + 1+1+3 + 1+ 3 + 1+ 4 + 4

rg + s = 5+ 4 + 1+ 4 + 1+ 3 + 1 + 1+ 3 + 1 + 1+ 2 + 1 + 1 + 1 +
+ 2 + 8(1) + 1 + 8(1) + 2 + 1+1 + 1+ 2 + 1 + 1+ 3 + 1 +
+1+3+1+4+1+4+5

r 9 + 7 = 7 + 2 + 5 + 2 + 2 + 3 + 2 + 2 + 2 + 1+ 2 + 2 + 3 + 1 + 1 +
+2+3+2+1+1+3+2+2+1+2+2+2+3+2+
+2+5+2+7

19^8 = 8 + 1 + 7 + 1 + 1+ 6 + 1+ 2+ 5 + 1+ 3 + 4 + 1+ 4 + 3 +
+1+4+3+1+5+2+1+6+1+1+7+1+8

r\r\cs\f> Original from

Digitized by ^UUglL UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS 277

C. Pitch Scales 2

In the field of music, our equal temperament system of tuning constitutes
the primary selective system. Equal temperament tuning involves 12 semitone
units — c, c#, d, d#, e, f, f#, g, g#, a, a#, b and c. These units became established
over a period of centuries, as reference points among all the possible frequencies
that constitute the audible continuum.

Tuning is a problem of pitch, and pitch is a matter of frequency. In equal
temperament tuning, the twelve tonal units are so related that the second c in
the series above is an octave higher than the first c — i.e., its frequency is twice
the frequency of the first c. The intervening units comprise a series of uniform
ratios, which are related as logarithms to the base \fT.

The entire series takes the following form:

c

2 tV =

1

c

2 -h =

D

2* =

V2~

n

3

l. —

-vVT

E

2* =

F

2* =

F

2*-

vT

G

2* =

G

2* =

V?

A

2* =

</¥

A

2 # =

</¥

B

n

2 17 =

C

2&

2

From this tuning system, certain sequences of tones, or scales, may be
abstracted. These constitute secondary selective systems and provide the raw
material for musical composition. A summary of the pitch scales possible in
our tuning system follows.

*A comprehensive analysis of pitch scales will be
found in Book II, Theory of Pilch Scales of The
Schillinger System of Musical Composition. (EH.)

r\r\n\f> Original from

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278

TECHNOLOGY OF ART PRODUCTION

/. Number of Pitch Scales in all Groups of One Axis 3

Group I 2048

Group II 9217

Group III 48

Group IV 2000

Grand Total 13,313

2. Number of Pitch Scales in All Four Groups
of Equal Temperament {\/Y )

The First Group:
The Second Group:
The Third Group:
The Fourth Group:

2" = 2048 scales
9217
48 scales

2 7 + 2 8 + 2 9 + 2 4 + 2 6 + 2 10 =

+ 128 + 256 + 512 + 16 + 64 + 1024 =2000
3. The Number of Pitch Scales in the First Group.

1 unit

(P)

1

2 units

(2p)

11

3 units

(3p)

55

4 units

(4p)

165

5 units

(5p)

330

6 units

(6p)

462

7 units

(7p)

462

8 units

(8p)

330

9 units

(9p)

165

10 units (lOp)

55

11 units (lip)

11

12 units (12p)

1

2" =

2048

•Schillinger does not follow the traditional sys-
tem of classifying scales simply as major, minor
and chromatic, for these classifications arc pat-
ently not broad enough to encompass all possible
scales, or to embrace even those modern scale
forms that have, in recent years, become common-
places in our musical vocabulary. The four large
groups into which Schillinger divides pitch scales
may be described as follows. Group I: Assymctric
scales with on root-tone and a range of one
octave. This group includes the seven-unit dia-
tonic scales (known as "major" and "minor"),
which constitute our traditional musical language
and serve as the basis of traditional harmony.

Digitized byGoOgle

Group II: Expanded scales with one tonic and a
range of more than one octave. These are obtained
by tonally expanding the scales of the first group,
i.e., by rearranging the mutual positions of the
specific pitch-units. Group III: Symmetric scales
with more than one root-tone and a range of one
octave. Derived from roots of the number 2,
these scales are based on pitch-units which con-
stitute the symmetric points between the root-
tones. Group IV: Symmetrical scales with more
than one root-tone and a range exceeding an
octave. Like the symmetric scales in Group III,
these scales contain an equal number of semitones
between pitch-units. (Ed.)

Original from
UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

279

GROUP I. ONE ROOT-TONE. RANGE: 11

i j j ij j M

Two-unit scales. One interval.
Number of scales: 11

a + a

a + «

Three-unit scales. Two intervals.
Number of scales: 55

p a +a + i

3 + a + * ,

i jjjji

Four-unit scales. Three intervals.
Number of scales: 165

a+ a + i +•(§)+■ 2 » a ■>• i
2

Seven-unit scales. Six intervals.
Number of scales: 462

Figure 1. Group I. Pitch-Scales.

4. The Number of Pitch-Scales in the Second Group

3p

K

X

55

55 scales

4p

2 E

X

165

330 scales

5p

3 e

X

330

990 scales

6p

4 e

X

462

1848 scales

7p

5k

X

462

2310 scales

8p

6 E

X

330

1980 scales

9p

7 B

X

165

1 155 scales

lOp

«e

X

55

440 scales

lip

9 E

X

11

99 scales

12p

10e

X

1

10 scales

Digitized by GoOgle

9217

Original from
UNIVERSITY OF MICHIGAN

280

TECHNOLOGY OF ART PRODUCTION

GROUP II. ONE ROOT-TONE. RANGE: OVER 12

Eo = Zero expansion

XJT

II

JU H T 71 m

Ei = First expansion

HI

y zn n it vi

E 2 = Second expansion «t

XE

17 VTT TTT VT TT

Ej = Third expansion

E4 = Fourth expa nsion —

IT I 1 7 I

Digiti

E 6 = Fifth Expansion

m zr y is hj

Figure 2. Pitch-Scales of Group II.

Cnr%ci\{> Original from

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SELECTIVE SYSTEMS

281

5 . The Third Group of Pitch-Scales {Symmetric System)

(1) 1 (4) \/l = 3

(2) VT # = 6 (5) </2 ^ = 2

(3) </l & = 4 (6) ^7 # = 1

GROUP III. MORE THAN ONE ROOT-TONE. RANGE: 12

Two tonics:
6 semitones between tonics

Two-Unit Scale

Three-Unit Scale

\, 9 »" m

EE

Three tonics:
4 semitones between tonics

Two-Unit Scale

Three-Unit Scale

Four tonics:
3 Semitones between tonics

3 Semitones between tonics

Two-Unit Scale

? , t»o 0'

Six tonics: 2 semitones between tonics

4

Two-Unit Scale

1

Twelve tonics: 1 semitone between tonics

Figure 3. Pitch-Scales of Group III.

rv -■ _j i- frinnlf* Original from

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282

TECHNOLOGY OF ART PRODUCTION

6. The Fourth Group of Pitch-Scales (Symmetric System)

(1)

*-

8

(4)

^32

5

(2)

<K

^ =

9

(5)

7

(3)

</T2

10

(6)

-^2048

# -

11

GROUP IV. MORE THAN ONE ROOT-TONE
RANGES: 24, 36, 60, 132

Three tonics:
8 semitones between roots

All sectional scales of fourth group,
starting from their symmetrical points,
have identical construction.

•

Four tonics: 9 semitones between roots

Five units

Six tonics:
10 semitones between rootSj

Twelve tonics:
1 1 semitones between roots #

XX

The final C is 11 octaves above beginning C.

Digits

Figure 4. Pitch-Scales of ^ rou ^Y\W

m

UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS
7 . Symmetric Scales of Equal Temperament

283

Group

Range of
Symmetry

Interval of
Symmetry

Number
of Tonics

Number
of Scales

Value

III.

1

1

No. of
Semitones

1

1

1

III.

2

V

4f= i

12

1

1

III.

2

V

¥-= 2

6

2

2

III.

2

v **

■¥-= 3

4

4

2*

III.

2

V

Jf = 4

3

8

2 s

IV.

32

ty32

ft = 5

12

16

2*

III.

2

V

^ = 6

2

32

2 s

IV.

128

ft = 7

12

64

2 4

IV.

4

* = 8

3

128

2 7

IV.

8

¥ = 9

4

256

2 8

IV.

32

^32

T = 10

6

512

2 9

IV.

2048

-^2048

W"= 11

12

1024

2io

Figure 5. Summary Analysis of Symmetric Scales.

Six forms of symmetry in group III., and six forms of symmetry in group IV.

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284 TECHNOLOGY OF ART PRODUCTION

D. Scales of Linear Configuration and Area. 4

/ . Periodicity of Dimensions .

A. Monomial Periodicity of Rectilinear Segments Moving Under a
Constant Angle.

a. Rectilinear Segments.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°. (See Figure 1.)

Figure 1. Rectilinear segments moving under a constant angle.

4 Student* will find that tome of these scales are
illustrated. It was Schillinger's intention that the
student work out the others as exercises. (Ed.)

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SELECTIVE SYSTEMS

285

b. 180° Arcs Moving in a Constant Clockwise Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°. (See Figure 2.)

286

TECHNOLOGY OF ART PRODUCTION

c. 180° Arcs Moving in a Constant Alternating Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°. 330°,
360°. (See Figure 3.)

Figure 3. 180° arcs moving in a constant alternating direction.

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SELECTIVE SYSTEMS

287

B. Binomial Periodicity of Rectilinear Segments Moving Under a
Constant Angle. (2 + 1) + . . .

a. Rectilinear Segments.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

b. 180° Arcs Moving in a Constant Clockwise Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

c. 180° Arcs Moving in a Constant Alternating Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

C. Binomial Periodicity of Rectilinear Segments Moving Under a
Constant Angle. (3 + 1) + • • •

a. Rectilinear Segments.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

b. 180° Arcs Moving in a Constant Clockwise Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

c. 180° Arcs Moving in a Constant Alternating Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

D. Binomial Periodicity of Rectilinear Segments Moving Under a
Constant Angle. (3 + 2) 4- . . .

a. Rectilinear Segments.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

b. 180° Arcs Moving in a Constant Clockwise Direction.

1 . 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°, 360°.

c. 180° Arcs Moving in a Constant Alternating Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°.
360°.

E. Binomial Periodicity of Rectilinear Segments Moving Under a
Constant Angle. (4 + 3) + . . .

a. Rectilinear Segments.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

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288

TECHNOLOGY OF ART PRODUCTION

b. 180° Arcs Moving in a Constant Clockwise Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

c. 180° Arcs Moving in a Constant Alternating Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

F. Trinomial Periodicity of Rectilinear Segments Moving Under a
Constant Angle. (3 -f 2 + 1) + . . .

a. Rectilinear Segments.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

b. 180° Arcs Moving in a Constant Clockwise Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

c. 180° Arcs Moving in a Constant Alternating Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

INFINITE SERIES

G. Infinite Series of Rectilinear Segments Moving Under a Constant
Angle— Series 1:1+2 + 3 + 4 + 5 + 6+ . . .

a. Rectilinear Segments.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

b. 180° Arcs Moving in a Constant Clockwise Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

c. 180° Arcs Moving in a Constant Alternating Direction.

1. 0°, 30°, 60°, 90°, 120°, 150°, 180°, 210°, 240°, 270°, 300°, 330°,
360°.

H. Infinite Series of Rectilinear Segments Moving Under a Constant
Angle— Series 11:1+3 + 5 + 7 + 9+ . . .

a. Rectilinear Segments.

1. 0°, 30°, 60°, 90°, 120°.

2. 150°, 180°, 210°.

3. 240°, 270°, 300°, 330°, 360°.

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SELECTIVE SYSTEMS

289

b. 180° Arcs Moving in a Constant Clockwise Direction.

1. 0°, 30°, 60°, 90°, 120°.

2. 150°, 180°, 210°.

3. 240°, 270°, 300°, 330°, 360°.

c. 180° Arcs Moving in a Constant Alternating Direction.

1. 0°, 30°, 60°, 90°, 120°.

2. 150°, 180°, 210°.

3. 240°, 270°, 300°, 330°, 360°.

I. Infinite Series of Rectilinear Segments Moving Under a Constant
Angle— Series III: 1+2 + 3 + 5 + 8 + 13+ . . .

a. Rectilinear Segments.

1. 0°, 30°, 60°, 90°, 120°.

2. 150°, 180°, 210°.

3. 240°, 270°, 300°, 330°, 360°.

b. 180° Arcs Moving in a Constant Clockwise Direction.

1. 0°, 30°, 60°, 90°, 120°.

2. 150°, 180°, 210°.

3. 240°, 270°, 300°, 330°, 360°.

c. 180° Arcs Moving in a Constant Alternating Direction.

1. 0°, 30°. 60°, 90°, 120°.

2. 150°, 180°, 210°.

3. 240°, 270°, 300°, 330°, 360°.

J. Infinite Series of Rectilinear Segments Moving Under a Constant
Angle— Series IV: 1 + 2 + 4 + 7 + 11 + 16 + . . .

a. Rectilinear Segments.

1. 0°, 30°, 60°, 90°, 120°.

2. 150°, 180°, 210°.

3. 240°, 270°, 300°, 330°, 360°.

b. 180° Arcs Moving in a Constant Clockwise Direction.

1. 0°, 30°, 60°, 90°, 120°.

2. 150°, 180°, 210°.

3. 240°, 270°.

4. 300°, 330°, 360°.

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TECHNOLOGY OF ART PRODUCTION

c. 180° Arcs Moving in a Constant Alternating Direction.

1. 0°, 30°, 60°, 90°, 120°.

2. 150°, 180°, 210°.

3. 240°, 270°.

4. 300°, 330°, 360°.

K. Infinite Series of Rectilinear Segments Moving Under a Constant
90° Angle.

a. Rectilinear Segments.

1 . Series I , Series 1 1 (See Figure 4) .

2. Series III, Series IV.

1, 2, 3, 4, 5 17 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25

Figure 4. Infinite series of rectilinear segments moving under a constant angle.

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SELECTIVE SYSTEMS

b. 180° Arcs Moving in a Constant Clockwise Direction.

1. Series I, Series II (See Figure 5).

2. Series III, Series IV.

291

292

TECHNOLOGY OF ART PRODUCTION

c. 180° Arcs Moving in a Constant Alternating Direction.

1. Series I, Series II (See Figure 6).

2. Series III, Series IV.

1,2,3,4,5,6,7 15 1,3,5,7,9 21

Figure 6. 180° arcs moving in a constant alternating direction.

L. 180° Arcs Moving in a Constant Clockwise Direction, Using 180°
As Diameters.

1 . Series I , Series 1 1 .

2. Series III, Series IV.

rv -■ nr\cs\{> Original from

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SELECTIVE SYSTEMS

293

Z . Periodicity of A ngles .

A. Monomial Periodicity of Angles, with Constant Dimensions of
Rectilinear Segments.

a. Rectilinear Segments,

1. 10°, 20°, 30°, 40°, 60°, 90°. (See Figure 7.)

10'

10'

HO*

60"

30*

Figure 7, Monomial periodicity of angles, with constant dimensions

of rectilinear segments.

n . .. r\r\cs\(> Original from

Digitized by VjUUJJU. UNIVERSITY OF MICHIGAN

294

TECHNOLOGY OF ART PRODUCTION

b. 180° Arcs Moving in a Constant Clockwise Direction.
(Rectilinear Segments used as Diameters.)
1. 20°, 30°, 40°, 60°, 90°. (See Figure 8.)

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SELECTIVE SYSTEMS

c. 180° Arcs Moving in a Constant Alternating Direction,
(Rectilinear Segments Used as Diameters.)
I. 20°, 30°, 40°, 60°, 90°. (See Figure 9.)

30*

to'

9o'

Figure 9. 180° arcs moving in a constant alternating direction.

d. Rectilinear Segments.
1. 120*. 140°, 150°.

e. 180° Arcs Moving in a Constant Clockwise Direction.
1. 120°, 140°, 150°.

f- 180° Arcs Moving in a Constant Alternating Direction.
I. 120°, 140°, 150°.

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TECHNOLOGY OF ART PRODUCTION

g. Rectilinear Segments.

1. 150°, 160°, 170°. (See Figure 10.)

iro* no*

Figure 10. Monomial periodicity of angles with constant dimensions

of rectilinear segments .

h. 180° Arcs Moving in a Constant Clockwise Direction.
1. 150°, 160°, 170°. (See Figure 11.)

is-o* , jo-

Figure 11. 180° arcs moving in a constant clockwise direction.

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SELECTIVE SYSTEMS

297

B. Binomial Periodicity of Angles, with Constant Dimensions of
Rectilinear Segments.

a. Rectilinear Segments.

1. (10° + 20°) + . . . (10° + 30°) + . . . (20° + 40°) + . . .
. . . + (30° + 60°) + . . . (30° + 90°) + ... (See Figure 13.)

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TECHNOLOGY OF ART PRODUCTION

SELECTIVE SYSTEMS 299

180° Area Moving in a Constant Clockwise Direction,
(Rectilinear Segments Used as Diameters.)

1. (10° + 20°)+ , . . (10° + 30°)+ . . . (20* + 40°) + . . .
. . . + (30° + 60°) + ... (30° + 90°) + ... (See Figure 14.)

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TECHNOLOGY OF ART PRODUCTION

c. 180° Arcs Moving in a Constant Alternating Direction.
(Rectilinear Segments Used as Diameters,)

1. (10° + 20°) + . . . (10° + 30") + . . . (20° -f 40°) + . . .
.". . + (30° + 60°) + . . . (30° + 90°) + . . , (See Figure 15.)

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SELECTIVE SYSTEMS

301

C. Trinomial Periodicity of Angles.

a. Rectilinear Segments; 180° Arcs Moving in a Constant Counter-
Clockwise Direction; 180° Arcs Moving in a Constant Clockwise Direc-
tion. 1+2 + 6.

1. 10° + 20° + 60°. (See Figure 16.)

302

TECHNOLOGY OF ART PRODUCTION

b. Rectilinear Segments. 1 + 2 + 3.

1. (10° + 20° + 30°) + • • ■ (15° + 30° + 45°) + . . .
. . . + (18° + 36° + 54°) + . . .

2. (20° + 40° + 60°) + . . . (30° + 60° + 90°) + . . .

c. 180° Arcs Moving in a Constant Clockwise Direction. 1+2-1-3.

1. (10° + 20° + 30°) + . . . (15° + 30° + 45°) + . . .
. . .'+ (18° -|- 36° + 54°) + . . .

2. (20° + 40° + 60°) + . . . (30° 4- 60° + 90°) + . . .

d. 180° Arcs Moving in a Constant Alternating Direction. 1+2 + 3.

1. (10° + 20° + 30°) + . . . (15° + 30° + 45°) + . . .
. . . + (18° + 36° + 54°) + . . .

2. (20° + 40° + 60°) + . . . (30° + 60° + 90°) + . . .

3 . Rectilinear Segments Forming A ngles in A Iternating Directions

A. Rectilinear Segments.

a. Binomials by the Sum.

1. S = 30°, 5 + 1. 4+1,3 + 1,2 + 1, 3 + 2 (See Figure 17.)

2. S = 60°, 5 + 1,4 + 1,3 + 1,2 + 1,3 + 2, 5 + 4 (See Figure 18.)

3. S = 90°, 5 + 1, 3 + 1, 2 + 1, 3 + 2, 5 + 4

4. S = 120°, 4 + 1, 3 + 1, 2 + 1, 3 + 2, 7 + 5

5. S = 150°, 4 + 1,3 + 2,8 + 7

6. S = 180°, 5 + 1, 3 + 1, 2 + 1, 11 + 7, 3 + 2. 5 + 4
(See Figure 19.)

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TECHNOLOGY OF ART PRODUCTION

SELECTIVE SYSTEMS

305

b. Binomials by the Initial Unit.

1.

10° =

1,7 + 1,5 + 1,4 + 1,3 + 1,2 + 1

2.

10° =

1, 5 + 2,3 + 2, 7 + 3, 5 + 3, 4 + 3

3.

10°-

1,7 + 4,5 + 4

4.

10°-

1, 8 + 5, 7 + 5, 6 + 5

5.

10°-

1, 7 + 6, 11 + 7.9 + 7

6.

10°-

1,8 + 7

7.

10 8 -

1,9 + 8

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B. ISO Arcs Moving in a Constant Direction.

(Rectilinear Segments Serve as Diameters.)

a. Binomials by the Sum.

1 . S = 30°, 5 + 1,4 + 1,3 + 1,2 + 1,3 + 2 (See Figure 20.)
2.S= 60", B + 1,4 + 1,3 + 1,2 + 1,3 + 2,5 + 4
(See Figure 21.)

3. S = 90°, 5 + 1, 3 + 1, 2 + 1,3 + 2, 5 + 4

4. S - 120°, 4 + 1,3 + 1, 2 + 1,3 + 2, 7 + 5

5. S = 150°, 4 + 1,3 + 2,8 + 7

6. S = 180°, 5 + 1, 3 + 1, 2 + 1, 11 + 7, 3 + 2, 5 + 4
(See Figure 22.)

Figure 20. 1 80° arcs moving in a constant direction. S = 3Q°

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Binomials by the Initial Unit.

1.7 + 1,5 + 1,4 + 1,3 + 1,2 + 1
1, 5 + 2,3 + 2, 7 + 3, 5 + 3,4 + 3
1,7+4,5+4
1, 8 + 5. 7 + 5, 6 + 5
1, 7 + 6, 11 + 7,9 + 7

1.8 + 7

1.

10°

2.

10°

3.

10°

4.

10°

5.

10°

6.

10°

7.

10°

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SELECTIVE SYSTEMS

309

C. 180° Arcs Moving in a Constant Alternating Direction.

a. Binomials by the Sum.

L, S - 30°, 5 + I, 4 + 1, 3 + 1, 2 + 1, 3 + 2 (See Figure 23.)

2. S = 60°, 5 + 1,4 + 1,3+1,2 + 1,3 + 2,5 + 4 (See Figure 24.)

3. S - 90°, 5 + 1,3 + 1,2 + 1,3 + 2 r S +4

4. S = 120°, 4 + 1,3 + I, 2 + 1,3 + 2, 7+5

5. S = 150°, 4 + 1, 3 + 2,8 + 7

6. S = 180°, 5 + 1, 3 + 1, 2 + 1, 11 +7, 3 + 2, 5+4
(See Figure 25.)

Figure 23. 180° arcs moving in constant alternating directions. S~ 30°

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TECHNOLOGY OF ART PRODUCTION

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TECHNOLOGY OF ART PRODUCTION

b. Binomials by the Initial Unit.

1. 10°= 1,7 + 1,5 + 1,4 + 1,3 + 1,2 + 1

2. 10°= 1,5 + 2,3 + 2,7 + 3,5 + 3,4 + 3

3. 10°= 1, 7 + 4, 5 + 4

4. 10°= 1, 8 + 5, 7 + 5, 6 + 5

5. 10°= 1, 7 + 6, 11 + 7, 9 + 7

6. 10°= 1,8 + 7

7. 10°= 1,9 + 8

D. Trinomial Periodicity of Angles Moving in Alternating Direction.
(Rectilinear Segments Serve as Diameters.)

a. Rectilinear Segments — Arcs Move in One Direction and in Alternating
Direction.

1. 1-^2 + 3, 15° + 30° + 45°, Rectilinear Segments, 180° Arcs moving
in a constant direction, and 180° arcs moving in a constant alter-
nating direction. 30° + 60° + 90°.

2. 3-^4-^5, 15° + 20° + 25°, 30° + 40° + 50°, 60° + 80° + 100°.
Rectilinear Segments, 180° arcs moving in a constant direction, and
180° arcs moving in a constant alternating direction.

£. Symmetric Construction of Angles Moving in Alternating Direction
Under Infinite Series.

a. Rectilinear Segments.
1. Series I, II, III, IV.

b. 180° Arcs Moving in a Constant Clockwise Direction (Rectilinear
Segments Serve as Diameters) .

1. Series I, II, III, IV.

c. 180° Arcs Moving in a Constant Alternating Direction. (Rectilinear
Segments Serve as Diameters).

1. Series I, II, III, IV.

F. Symmetric Construction of Rectilinear Segments with Respect to
their Dimensions and Periodicity of Angles Following Infinite
Series, with Alternating Direction.

a. Rectilinear Segments.

1 . Series I and II.

2. Series III and IV.

b. 180° Arcs Moving in a Constant Clockwise Direction. (Rectilinear
Segments Serve as Diameters).

1. Series I and II.

2. Series III and IV.

Digitized byGoOgle

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SELECTIVE SYSTEMS

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c. 180° Arcs Moving in a Constant Alternating Direction (Rectilinear
Segments Serve as Diameters) .

1. Series I and II.

2. Series III and IV.

4. Monomial, Binomial and Trinomial Periodicity of Sector Radii.

A. Discontinuous Counter-Clockwise 180° Arcs.

180° Arcs, 10° Sectors; 30° Sectors; 30° + 60° Sectors; 30° + 60° + 90°
Sectors.

B. Discontinuous 180° Arcs in Alternating Direction.

180° Arcs, 30° Sectors; 30° + 60° Sectors; 30° + 60° + 90° Sectors.

C. Infinite Series of the Sector Radii, Angle Variation of Values,
Counter-clockwise 180° discontinuous Arcs.

Series I, II, III, IV.

D. Discontinuous 180° Arcs in Alternating Direction.

Series I, II, III, IV. (See Figure 26.)

Figure 26. Discontinuous 180° arcs moving in alternating direction.

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TECHNOLOGY OF ART PRODUCTION

5. Periodicity of Radii and Angle Values.

A. Arcs Moving in a Constant Direction.

1. 90° Arc, Constant Clockwise Direction.
S r - 1,2,3,4, 5,6 ... .

2. 90° Arc, Constant Clockwise Direction,.
S r - 1,2,3,5,8, 13 ... .

3. 90° Arc, Constant Clockwise Direction.
S, - 1, 2, 4, 7, 11, 16, 22, 29 ... .

4. 150° Arc, Constant Clockwise Direction.
Sr - 1, 2, 3, 4, 5, 6 ... .

5. 150° Arc, Constant Clockwise Direction.
S r - 1, 2, 3, 5, 8, 13, 21 ... .

6. 150° Arc, Constant Clockwise Direction.
Sr - 1, 2,4, 7, 11, 16, 22 ... .

7. 240° Arc, Constant Clockwise Direction.
S r - 1, 2, 3 . . . .

8. 240° Arc, Constant Clockwise Direction.
* Sr - 1, 2, 3, 5, 8, 13, 21 ... .

9. 240° Arc, Constant Clockwise Direction.
Sr - 1, 2, 4, 7, 11, 16, 22 ... .

10. 270° Arc, Constant Clockwise Direction.

Sr - 1, 2, 3, 4, 5, 6 ... . (See Figure 27.)

11. 270° Arc, Constant Clockwise Direction.
Sr - 1, 2, 3, 5, 8, 13, 21 ... .

12. 270° Arc, Constant Clockwise Direction.
Sr - 1, 2, 4, 7, 11, 16, 22 ... .

13. 330° Arc, Constant Clockwise Direction.

S r - 1, 2, 3, 4, 5, 6 ... . (See Figure 28.)

14. 330° Arc, Constant Clockwise Direction.
Sr - 1, 2, 3, 5, 8, 13, 21 ... .

15. 330° Arc, Constant Clockwise Direction.
Sr - 1, 2, 4, 7, 11, 16, 22 ... .

rv -■ f\r\Ci\{> Original from

ngitizeti by o UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS 315

16. 360° Arc, Constant Clockwise Direction.
S r ■« 1 , 2, 3, 4, 5, 6 . . . .

17. 360° Arc, Constant Clockwise Direction.
S f - 1, 2, 3, 5, 8, 13, 21 ... .

18 . 360° Arc, Constant Clockwise Direction.
S, - 1, 2, 4, 7, 11, 16, 22 ... .

Ratios of t 3t

y

Figure 27. 270° arc moving in a constant clockwise direction.

Digitized by LiOOglC UNIVERSITY OF MICHIGAN

Figure 28. 330° arc moving in a constant clockwise direction.

rv in _j i» r\rscs\f> Original from

Digitized by VjUUJJU. UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

317

B. Arcs Moving in a Constant Alternating Direction.

(Sin and cos.)

1. 90° Arc, Constant Alternating Direction.
S, = 1, 2,3, 4, 5, 6 ... .

2. 90° Arc, Constant Alternating Direction.
S r = 1, 2, 3, 5, 8, 13 ... .

3. 90° Arc, Constant Alternating Direction.
S r = 1, 2, 4, 7, 11, 16, 22, 29 ... .

4. 150° Arc, Constant Alternating Direction.
S r = 1, 2, 3, 4, 5, 6 . . . .

5. 150° Arc, Constant Alternating Direction.
S r = 1, 2, 3, 5, 8, 13, 21 ... .

6. 150° Arc, Constant Alternating Direction.
S r = 1, 2, 4, 7, 11, 16, 22 ... .

7. 240° Arc, Constant Alternating Direction.
S r = 1, 2, 3 . . . .

8. 240° Arc, Constant Alternating Direction.
S r = 1, 2, 3, 5, 8, 13, 21 ... .

9. 240° Arc, Constant Alternating Direction.
S r = 1, 2, 4, 7, 11, 16, 22 ... .

10. 270° Arc, Constant Alternating Direction.
S r = 1, 2, 3, 4, 5, 6 . . . .

11. 270° Arc, Constant Alternating Direction.
S r = 1, 2, 3, 5, 8, 13, 21 ... .

12. 270° Arc, Constant Alternating Direction.
S r = 1, 2, 4, 7, 11, 16, 22 ... .

13. 330° Arc, Constant Alternating Direction.
S r = 1, 2, 3, 4, 5, 6 ... .

14. 330° Arc, Constant Alternating Direction.
S r = 1, 2, 3, 5, 8, 13, 21 ... .

15. 330° Arc, Constant Alternating Direction.
S r = 1, 2, 4, 7, 11, 16, 22 ... .

16. 360° Arc, Constant Alternating Direction.
S r = l,2,3,4,5,6. .. .

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TECHNOLOGY OF ART PRODUCTION

17. 360° Arc, Constant Alternating Direction.
S r = 1, 2, 3, 5, 8, 13, 21 ... .

18. 360° Arc, Constant Alternating Direction.
S r = 1, 2, 4, 7, 11, 16, 22 ... .

19. Derivative Design. (See Figure 29.)

20. Derivative Design. (See Figure 30.)

Digi

Figure 29

Google

Derivative design.

Original from

UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

319

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TECHNOLOGY OF ART PRODUCTION

C. Alternating Sin Movement of Arcs.

1. 90° Arc, Alternating Sin Movement of Arcs.
S r = 1 , 2, 3, 4, 5, 6 . . . . (See Figure 31.)

2. 90° Arc, Alternating Sin Movement of Arcs.
S r = 1,2, 3, 5, 8, 13 . . . .

3. 90° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 4, 7, 11, 16, 22, 29 ... .

4. 150° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 3. 4, 5, 6 . . . .

5. 150° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 3, 5, 8, 13, 21 ... .

6. 150° Arc, Alternating Sin Movement of Arcs.

S r = 1, 2, 4, 7, 11, 16, 22 ... . (See Figure 32.)

7. 240° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 3 . . . .

8. 240° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 3, 5, 8, 13, 21 ... .

9. 240° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 4, 7, 11, 16, 22 ... .

10. 270° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 3, 4, 5, 6 . . .

11 . 270° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 3, 5, 8, 13, 21 ... .

12. 270° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 4, 7, 11, 16, 22 ... .

13. 330° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 3, 4, 5, 6 . . . .

14. 330° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 3, 5, 8, 13, 21 ... .

15. 330° Arc, Alternating Sin Movement of Arcs.
S r = 1, 2, 4, 7, 11, 16, 22 ... .

16. Derivative Design. (See Figure 33.)

17. Derivative Design. (See Figure 34.)

18. Derivative Design. (See Figure 35.)

Original from
UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

321

Figure 31, 90° arcs moving in alternating sin movement of arcs.

Figure 32. 150" arcs moving in alternating sin movement of arcs.

Digitized by ^OOglC UNIVERSITY OF MICHIGAN

TECHNOLOGY OF ART PRODUCTION

r - 1, 2,4, 7, 11, 16

Figure 33. Derivative design.

j + T +T + T + T + T +2r

^ 45° + 90° + 135° + 180° + 225° + 270° + 315° + 360
r-i f 2,4.

Figure 34. Derivative design.

Original from
UNIVERSITY OF MICHIGAN

Google

SELECTIVE SYSTEMS

323

324

TECHNOLOGY OF ART PRODUCTION

6. Periodicity of Arcs and Radii.

A. Variable Lengths of Arcs in Constant Clockwise Direction, and
Variable Radii.

1. 3-^-2, 120° + 60° + 60° + 120°
r 2 + 1 + 1 + 2

4- S-3, 90° + 30° + 60° + 60° + 30° + 90°
r3 + l+ 2 + 2 + l+ 3

5- H3, 72° + 48° + 24° + 72° + 24° + 48° + 72°
r3 + 2 + l+ 3 + l+ 2 + 3 (See Figure 36.)

2. Ellipses

2(30° + 60° + 90°)
r 2(1+2+3)

2(10° + 20° + 30° + 60° + 30° + 20° + 10°)
r 2(1 + 2 + 3 + 6 + 3 + 2 + 1)

3. (120° + 60° + 60° + 120°) (120° + 60° + 60° + . . . )
r (2 + 1 + 1 + 2) (2 + 1 + 1 + . . . . )

4. Variation of 4 4- 3 in Table Al above.

90° + 30° + 60° + 60° + 30° + 90° + 30° + 60° + 60° + 30° +
+ 90° + 90° + 60° + 60° + 30° + 90° + 90° + 30° + 60° + 30° +
+ 90° + 90° + 30° + 60° + 30° + 90° + 90° + 30° + 60° + 60° +
+ 90° + 90° + 30° + 60° + 60° + 30° + 90° + 30° + 60° + 60° +
+ 30° + 90°.

r3 + l+ 2 + 2 + l+ 3 + l+ 2 + 2 + l+ 3 + 3+ 2 + 2 +
+1+3+3+1+2+1+3+3+1+2+1+3+3+1+
+2+2+3+3+1+2+2+1+3+1+2+2+1+3

(See Figure 37.)

Digii

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Original from
UNIVERSITY OF MICHIGAN

326

TECHNOLOGY OF ART PRODUCTION

SELECTIVE SYSTEMS

327

B. Variable Lengths of Arcs in a Constant Alternating Direction and
Variable Radii.

1 . 10° + 20° + 30° + 60°
rl + 2 + 3 + 6

2. 80° + 50° + 30° + 20° + 10°
rl+2 + 3 + 5 + 8

3. 160° + 100° + 60° + 40° + 20°
rl+2 + 3 + 5 + 8 (See Figure 38.)

4. 10° + 20° + 30° + 50° + 80°
rl+2 + 3 + 5 + 8 (See Figure 39.)

C. Ratio of the Radii in Relation to the Scale of the Curvature of Arcs.

328

TECHNOLOGY OF ART PRODUCTION

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Original from
UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

329

7. r 4+3 Rhythmic Groups in Linear Design {Dimensions, Directions,
Angles, Constant and Variable).

A. Angle Unit = 30°.

a. Rectilinear Segments Moving in Alternating Directions.

1. r 4+3 applied to dimensions and angles.

2. Segments of Table al used as diameters.

r 4+3 applied to the length of arcs moving in constant clockwise
direction.

3. Segments of Table al used as Diameters.

r 4t .3 applied to the length of arcs moving in constant alternating
direction.

4. Alternating continuity under 90° angle from the groups represented
on Tables al, 2 and 3.

5. Closed continuity under 90° angle from the groups represented on
Tables al , 2 and 3.

b. Rectilinear Segments Moving in a Constant Clockwise Direction.

1. r 4+ 3 applied to dimensions and angles.

2. Segments of Table bl used as diameters.

3. Segments of Table bl used as diameters, r 4+J applied to the length
of arcs moving in a constant alternating direction.

4. Alternating continuity under 90° angle from the groups represented
on Tables bl , 2 and 3.

5. Closed continuity under 90° angle from the groups represented on
Tables bl , 2 and 3.

c. Constant Dimensions, Variable Angles.

1. Rectilinear Segments moving in a constant alternating direction.

2. 180° Arc moving in a constant clockwise direction through the seg-
ments of Table cl, used as diameters.

3. 180° arc moving in a constant alternating direction through the
segments of Table cl used as diameters.

4. Continuous alternating patterns under 90° angle from Tables cl,
c2 and c3.

d. Constant Dimensions, Variable Angles moving in a constant clockwise
direction .

1 . Rectilinear Segments moving in a constant clockwise direction . (See
Figure 40.)

Digitized byGoOgle

Original from
UNIVERSITY OF MICHIGAN

330 TECHNOLOGY OF ART PRODUCTION

2. 180° Arcs moving in a constant clockwise direction through the seg-
ments of Table dl used as diameters. (See Figure 41 .)

3. 180° Arcs moving in a constant alternating direction through the
segments of Table dl used as diameters. (See Figure 42.)

4. Continuous patterns under 180° angle from the Tables dl, 2 and 3.
(See Figure 43.)

5. Closed patterns under 90° angle from the Tables dl, 2 and 3.
(See Figure 44.)

SELECTIVE SYSTEMS 331

TECHNOLOGY OF ART PRODUCTION

Google

Original from
UNIVERSITY OF MICHIGAN

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

334 TECHNOLOGY OF ART PRODUCTION

3:1:2:2:1:3 (with one 90° angle omitted in each cycle.)

SELECTIVE SYSTEMS

335

r 4+3 Rhythmic Groups in Linear Design (Dimensions, Directions,
Angles, Constant and Variable). Angle Unit = 10°.

a. Rectilinear Segments moving in alternating direction.

1 . r 4+J applied to dimensions and angles.

2. Segments of Table al used as diameters.

r 4+3 applied to the length of arcs moving in a constant clockwise
direction.

3. Segments of Table al used as diameters.

r 4+3 applied to the length of arcs moving in a constant alternating
direction.

4. Alternating continuity under 30° angle from group represented in
Tables al , 2 and 3.

5. Closed continuity under 150° angle from group represented in
Tables al , 2 and 3.

b. Rectilinear Segments moving in a constant clockwise direction.

1. r 4+3 applied to dimensions and angles.

2. Segments of Table bl used as diameters.

r 4 + 3 applied to the length of arcs moving in a constant clockwise
direction.

3. Segments of Table bl used as diameters.

r *+3 applied to the length of arcs moving in a constant alternating
direction.

4. Closed continuity of the patterns of the Tables bl, 2 and 3, under
30° angle.

5. Closed continuity of the patterns of the Tables bl, 2 and 3, under
90° angle.

c. Rectilinear segments moving in a constant alternating direction. Con-

stant dimensions, variable angles.

1. Rectilinear segments moving in a constant alternating direction.

2. 180° Arcs moving in a constant clockwise direction, through the
segments of Table cl used as diameters.

3. 180° Arcs moving in a constant alternating direction, through the
segments of Table cl used as diameters.

4. Continuous patterns from the Tables cl, 2 and 3, under 30° angle.

5. Closed patterns from the Tables cl, 2 and 3, without repeating the
30° angle in the first term of each consecutive group.

Ul I ill I I '.I I II

UNIVERSITY OF MICHIGAN

336 TECHNOLOGY OF ART PRODUCTION

d. Constant dimensions, variable angles moving in a constant clockwise
direction .

1. Rectilinear segments mo zing in a constant alternating direction.

2. 180° Arcs moving in a constant clockwise direction through seg-
ments of Table cl used as diameters.

3. 180° Arcs moving in a constant alternating direction through seg-
ments of Table cl used as diameters.

4. Closed continuity of the patterns of Tables cl, 2 and 3 under
30° angle.

C. Variable Direction of Arcs.

1 . Direction of arcs varying with each term.
Angle Unit = 10°.

2. Direction of arcs variable with each term.
Angle Unit = 30°.

3. Direction and dimensions of arcs varying with each term.
Angle Unit = 30°.

8. Planes.

A. Rhythmic Centers.

d. 4 X 3 area

1. (* + *) 2

2. (* +
3- (4- + *)*

e. 7 X 6 area
1- (tV + tV) s

2. (■& + A) 5

Original from
UNIVERSITY OF MICHIGAN

a. 1 X 1 area

1. (£ + i) 2

2. (i + i) 3

3. (i + £) 4

b. 2 X 1 area

1. (f + £) 2

2. (£ + i) 3

3- (f + i) 4

c. 3 X 2 area

1. (f + %? (See Figure 45.)
2- (f + f) 3 (See Figure 46.)

3. (f + f) 4 (See Figure 47.)

Digitized byGoOgle

SELECTIVE SYSTEMS

337

Exterior Sides:

/3 2V _ 9, _6_ 6_ ±
\5 5/ 25 25 25 + 25

Areas:

9X6 6X6 6X4 9X4 = 54,36 24 36
25 25 25 25 " 25 25 25 25

3:2

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

Figure 45. Rhythmic center of 3X2 area.

rv m frinnlr* Original from

Digitized by VjUUJJU. UNIVERSITY OF MICHIGAN

TECHNOLOGY OF ART PRODUCTION

" (s+sy

Figure 46. Rhythmic centers of 3X2 area.

3:2

X

X

"X

S

X

"X

x.

^

X

X

X

X.

*x

X

X

V.

X

V.

X

X

X

X

" X

■x

X

X

"X

Figure 47. Rhythmic centers of 3X2 area.

Digitized by Google UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

339

B. Monomial Periodicity of the Sector Radii Produced from a Rhyth-
mic Center (Sector Angle = 30°).

1 . Continuous circular arcs moving in one direction, 2X1 area.

2. Continuous circular arcs moving in one direction, 3X2 area.

3. Continuous circular arcs moving in one direction, 4X3 area.

4. Discontinuous circular arcs moving in alternating direction, 2 X 1 area.

5. Discontinuous circular arcs moving in alternating direction, 3 X 2 area.

6. Discontinuous circular arcs moving in alternating direction, 4 X 3 area.

7. Discontinuous circular arcs moving in one direction, 2X1 area.

8. Discontinuous circular arcs moving in one direction, 3X2 area.

9. Discontinuous circular arcs moving in one direction, 4X3 area.

C. Rhythmic Groups Applied to Rectangular Area. Vertical and
Horizontal Circular Permutations.

a. Horizontal extension.

1. r 3+2 combined coordinates.

2. r 3+2 phase arrangement and vertical coincidence.

3- r 4+ 3 phase arrangement and vertical coincidence.

4- r 6+3 phase arrangement and vertical coincidence.

5. r B+4 phase arrangement and vertical coincidence. (See Figure 48.)

b. Vertical extension.

1. r 3+2 horizontal coincidence and phase arrangement.

2. r 4+ 3 horizontal coincidence and phase arrangement.

3. r 6+3 horizontal coincidence and phase arrangement.

4. r 8+4 horizontal coincidence and phase arrangement.

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TECHNOLOGY OF ART PRODUCTION

Phase arrangement

5:4

Vertical Coincidence

4+1+3+2+2+3+1+4
20

1+3+2+2+3+1+4+4
20

3+2+2+3+1+4+4+1
20

2+2+3+1+4+4+1+3
20

2+3+1+4+4+1+3+2
20

3+1+4+4+1+3+2+2
20

1+4+4+1+3+2+2+3
20

4+4+1+3+2+2+3+1
20

Figure 48. r 8+4 phase arrangement and vertical coincidence.

Digitized by LiOOglC UNIVERSITY OF MICHIGAN

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D. Automatic Continuity of Arcs in Rectangular Areas.

a. 4 X 3 area.

1. Arrangement of the radii from rhythmic center, (a, b, c, d, and
x, y, z radii) (See Figure 49.)

2. Radius a % 13 arcs.

3. Radius b f 51 arcs.

4. Radius b / 53 arcs. (See Figure 50.)

5. Radius c \ 16 arcs. (See Figure 51.)

6. Radius c \ 61 arcs.

7. Radius d y 17 arcs.

8. Radius d ^* 27 arcs.

9. Radius x •-» 22 arcs.

10. Radius x «-o 42 arcs.

11. Radius y \ 20 arcs.

12. Radius y f 40 arcs.

13. Radius z J 35 arcs.

14. Rectilinear arc derivatives.

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TECHNOLOGY OF ART PRODUCTION

9. Closed Polygons Conceived as Monomial Periodicity of Angles,
Dimensions and Directions.

A. Formula: t =

180° (T-2)

Where t is the value of an angle and T the number of angles and sides

in a polygon.

180° (3-2)

= 60 c

180° (4-2) 9Q0

180° (5-2)

= 108 c

180° (6-2)

= 120°

180° (7-2) = l2g 4°
7 7

UWL - ,35-
8

180° (9-2)

180° (10-2)
10

= 140 c

= 144 c

180° (11-2) , 3'

— n — = 147 n

180° (12-2)
12

= 150 c

180° (13-2) = 4 •
13 13

180° (14-2) .2°
14 154 7

180° (15-2)
15

= 156 s

Digitized by GoOgle

_ 180° (16-2)
16

180° (17-2)
17

180° (18-2)
18

180° (19-2)
19

180° (20-2)
20

_ 180° (21-2)
21

180° (22-2)
22

180° (23-2)
23

180° (24-2)
24

180° (25-2)
25

180° (26-2) _
26

180° (27-2) =
27

180° (28-2)
28

Original from
UNIVERSITY OF MICHIGAN

1°
157-
2

160°

1 °
I61 19

= 162'

6°
162-
7

= 163-

11

8°
23

165°
3°

= 165

2°

166 F3

2°
166-
3

1°
167 7

SELECTIVE SYSTEMS

180°(2»-2) .,.17° 180° (33-2) 1
f— 167- t _ 169-

. 180° (30-2) _ ' . 180° (34-2) _ £

30 34 17

180° (31-2) ^ o 12 180° (35-2) 5°

. 180° (32-2) _ 168 3° _ 180° (36-2) _

32 4 36

B. Kinetic Geometry.

Formulation of Closed Polygons

S = 4 (St 180° + 90°) square

S = 3 (St 180° + 60°) triangle

S = 5 (St 180° + 108°) pentagon

S - 6 (St 180° + 120°) hexagon

(any)

S = * 1 180° || mt 180° infinite rectilinear extension
S = s point (geometrical)

S = st point (physical)

In one system :

S = 3 (smt 180° + 60°) triangle
S = 4 (smt 180° + 90°) square
where mt is a given period of time moving under 180°.

C. Geometry

Polygonal Series

$ Triadic composition of triangles

Tetradic composition of squares, rectangles
$ Pentadic composition of pentagons
$ Hexadic composition of hexagons
■f Heptadic composition of heptagons
£ Octadic composition of octagons
$ Enneadic composition of enneagons
■J-fl- Decadic composition of decagons
S Polyadic composition of polygons

rv in _j i» r\rscs\f> Original from

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346 TECHNOLOGY OF ART PRODUCTION

E. Color Scales

Color origin is determined by the geometrical origin of the circumference.

Color axis is determined by identical colors on the opposite sides of the
circumference.

The angle of a color-axis depends on the number of color-sectors in the color
range distributed through a circle.

The distribution of color sectors may be either through a circle or an ellipse.

1. Full Spectrum Distributed in Bilateral Symmetry Through 360°

Twenty-four Sectors
Each Sector = 15°

Yellow Origin Yellow- Green Origin Green Origin

(Violet Axis) (Violet- Red Axis) (Red Axis)

Yellow

0°

Yellow-Green

0°

Green

0°

Yellow-Green

15°

Green

15°

Green-Blue

15°

Green

30°

Green-Blue

30°

Blue

30°

Green-Blue

45°

Blue

45°

Blue-Violet

45°

Blue

60°

Blue-Violet

60°

Violet

60°

Blue-Violet

75°

Violet

75°

Violet-Red

75°

Violet

90°

Violet-Red

90°

Red

90°

Violet-Red

105°

Red

105°

Red-Orange

105°

Red

120°

Red -Orange

120°

Orange

120°

Red-Orange

135°

Orange

135°

Orange-Yellow

135°

Orange

150°

Orange-Yellow

150°

Yellow

150°

Orange- Yellow

165°

Yellow

165°

Yellow-Green

165°

Yellow

180°

Yellow-Green

180°

Green

180°

Yellow-Green

195°

Green

195°

Green-Blue

195°

Green

210°

Green-Blue

210°

Blue

210°

Green-Blue

225°

Blue

225°

Blue-Violet

225°

Blue

240°

Blue- Violet

240°

Violet

240°

Blue- Violet

255°

Violet

255°

Violet-Red

255°

Violet

270°

Violet-Red

270°

Red

270°

Violet-Red

285°

Red

285°

Red-Orange

285°

Red

300°

Red-Orange

300°

Orange

300°

Red-Orange

315°

Orange

315°

Orange-Yellow

315°

Orange

330°

Orange-Yellow

330°

Yellow

330°

Orange- Yellow

345°

Yellow

345°

Yellow-Green

345°

(360°)

(360°)

(360°)

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS 347

Figure 52. Full spectrum in bilateral symmetry through 360° yellow origin.

Digitized by LjOOgie UNIVERSITY OF MICHIGAN

348

TECHNOLOGY OF ART PRODUCTION

Green-Blue Origin

Blue Origin

Blue-Violet Origin

(Red-Orange Axis)

(Orange A xis)

(Orange- Yellow Axis)

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Blue

195°

Rlue-Vinlpt

195°

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V 1V_*IC L

195°

Blue-Violet

210°

Violet

210°

Violet-Red

210°

Violet

225°

Violet-Red

225°

Red

225°

Violet- Red

240°

Red

240°

Red-Orange

240°

Red

255°

Red-Orange

255°

Orange

255°

Red-Orange

270°

Orange

270°

Orange- Yellow

270°

Orange

285°

Orange-Yellow

285°

Yellow

285°

Orange- Yellow

300°

Yellow

300°

Yellow- Green

300°

Yellow

315°

Yellow-Green

315°

Green

315°

Yellow-Green

330°

Green

330°

Green-Blue

330°

Green

345°

Green-Blue

345°

Blue

345°

(360°)

(360°)

(360°)

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

349

Violet Origin Violet-Red Origin Red Origin

( Yellow Axis) ( Yellow-Green Axis) {Green Axis)

Violet

0°

Violet- Red

0°

Red

0°

Violet-Red

15°

Red

15°

Red-Orange

15°

Red

30°

Red-Orange

30°

Orange

30°

Red-Orange

45°

Orange

45°

Orange-Yellow

45°

Orange

60°

Orange-Yellow

60°

Yellow

60°

Orange- Yellow

75°

Yellow

75°

Yellow-Green

75°

Yellow

90°

Yellow-Green

90°

Green

90°

Yellow-Green

105°

Green

105°

Green-Blue

105°

Green

120°

Green-Blue

120°

Blue

120°

Green-Blue

135°

Blue

135°

Blue-Violet

135°

Blue

150°

Blue- Violet

150°

Violet

150°

Blue- Violet

165°

Violet

165°

Violet- Red

165°

Violet

180°

Violet-Red

180°

Red

180°

Violet-Red

195°

Red

195°

Red -Orange

195°

Red

210°

Red -Orange

210°

Orange

210°

Red-Orange

225°

Orange

225°

Orange-Yellow

225°

Orange

240°

Orange- Yellow

240°

Yellow

240°

Orange-Yellow

255°

Yellow

255°

Yellow-Green

255°

Yellow

270°

Yellow-Green

270°

Green

270°

Yellow Green

285°

Green

285°

Green-Blue

285°

Green

300°

Green-Blue

300°

Blue

300°

Green-Blue

315°

Blue

315°

Blue- Violet

315°

Blue

330°

Blue-Violet

330°

Violet

330°

Blue- Violet

345°

Violet

345°

Violet-Red

345°

(360°)

(360°)

(360°)

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

350

TECHNOLOGY OF

ART

PRODUCTION

Red-Orange Origin

(Green-Blue Axis)

Red-Orange 0°

Orange 15°

Orange- Yellow 30°

Yellow 45°

Yellow-Green 60°

Green 75°

Green-Blue 90°

Blue 105°

Blue-Violet 120°

Violet 135°

Violet-Red 150°

Red 165°

Red-Orange 180°

Orange 195°

Orange- Yellow 210°

Yellow 225°

Yellow-Green 240°

Green 255°

Green-Blue 270°

Blue 285°

Blue-Violet 300°

Violet 315°

Violet-Red 330°

Red 345°

(360°)

Orange Origin

(Blue Axis)

Orange

0°

Orange-Yellow

15°

Yellow

30°

Yellow-Green

45°

Green

60°

Green-Blue

75°

Blue

90°

Blue-Violet

105°

Violet

120°

Violet-Red

135°

Red

150°

Red-Orange

165°

Orange

180°

Orange- Yellow

195°

Yellow

210°

Yellow-Green

225°

Green

240°

Green-Blue

255°

Blue

270°

Blue-Violet

285°

Violet

300°

Violet-Red

315°

Red

330°

Red-Orange

345°

(360°)

Orange- Yellow Origin

(Blue-Violet Axis)

Orange-Yellow

0°

Yellow

195°

Yellow

15°

Yellow-Green

210°

Yellow-Green

30°

Green

225°

Green

45°

Green-Blue

240°

Green-Blue

60°

Blue

255°

Blue

75°

Blue-Violet

270°

Blue-Violet

90°

Violet

285°

Violet

105°

Violet-Red

300°

Violet-Red

120°

Red

315°

Red

135°

Red-Orange

330°

Red-Orange

150°

Orange

345°

Orange

165°

(360°)

Orange- Yellow

180°

i r\r\a\i* Original from

igitized by \ t UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

351

2. The Scale of Twelve Hues

a. FULL SPECTRUM

Each Sector = 30°

Yellow Origin YellowGreen Origin Green Origin

Yellow

0°

Yellow- Green

0°

Green

0°

Yellow-Green

30°

Green

30°

Green-Blue

30°

Green

60°

Green-Blue

60°

Blue

60°

Green-Blue

90°

Blue

90°

Blue- Violet

90°

Blue

120°

Blue- Violet

120°

Violet

120°

Blue- Violet

150°

Violet

150°

Violet-Red

150°

Violet

180°

Violet-Red

180°

Red

180°

Violet-Red

210°

Red

210°

Red-Orange

210°

Red

240°

Red -Orange

240°

Orange

240°

Red -Orange

270°

Orange

270°

Orange- Yellow

270°

Orange

300°

Orange- Yellow

300°

Yellow

300°

Orange- Yellow

330°

Yellow

330°

Yellow-Green

330°

Green-Blue Origin

Blue Origin

Blue-Violet Origin

Green-Blue

0°

Blue

0°

Blue-Violet

0°

Blue

30°

Blue-Violet

30°

Violet

30°

Blue-Violet

60°

Violet

60°

Violet-Red

60°

Violet

90°

Violet-Red

90°

Red

90°

Violet-Red

120°

Red

120°

Red-Orange

120°

Red

150°

Red-Orange

150°

Orange

150°

Red-Orange

180°

Orange

180°

Orange- Yellow

180°

Orange

210°

Orange- Yellow

210°

Yellow

210°

Orange-Yellow

240°

Yellow

240°

Yellow-Green

240°

Yellow

270°

Yellow-Green

270°

Green

270°

Yellow-Green

300°

Green

300°

Green- Blue

300°

Green

330°

Green-Blue

330°

Blue

330°

Digitiz<

Google

Original from
UNIVERSITY OF MICHIGAN

352

TECHNOLOGY OF ART PRODUCTION

Violet Origin Violet-Red Origin Red Origin

Violet

0°

Violet-Red

0°

Red

0°

Violet-Red

30°

Red

30°

Red -Orange

30°

Red

60°

Red-Orange

60°

Orange

60°

Red -Orange

90°

Orange

90°

Orange-Yellow

90°

Orange

120°

Orange- Yellow

120°

Yellow

120°

Orange -Yellow

150°

Yellow

150°

Yellow-Green

150°

Yellow

180°

Yellow-Green

180°

Green

180°

Yellow-Green

210°

Green

210°

Green-Blue

210°

Green

240°

Green-Blue

240°

Blue

240°

Green- Blue

270°

Blue

270°

Blue-Violet

270°

Blue

300°

Blue-Violet

300°

Violet

300°

Blue-Violet

330°

Violet

330°

Violet-Red

330°

Red -Orange Origin

Red-Orange 0°

Orange 30°

Orange- Yellow 60°

Yellow 90°

Yellow-Green 120°

Green 150°

Green-Blue 180°

Blue 210°

Blue-Violet 240°

Violet 270°

Violet-Red 300°

Red 330°

Orange Origin

Orange 0°

Orange-Yellow 30°

Yellow 60°

Yellow-Green 90°

Green 120°

Green-Blue 150°

Blue 180°

Blue-Violet 210°

Violet 240°

Violet- Red 270°

Red 300°

Red-Orange 330°

Orange- Yellow Origin

Orange-Yellow 0°

Yellow 30°

Yellow-Green 60°

Green 90°

Green-Blue 120°

Blue 150°

Blue-Violet 180°

Violet 210°

Violet-Red 240°

Red 270°

Red-Orange 300°

Orange 330°

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

353

b. 180° OF THE SPECTRUM DISTRIBUTED
THROUGH 360°.

Yellow Origin Yellow-Green Origin Green Origin

Green-Blue A xis

Blue Axis

Blue-Violet Axis

Yellow

0°

Yellow-Green

0°

Green

0°

Yellow-Green

30°

Green

30°

Green-Blue

30°

Green

60°

Green-Blue

ou

Blue

60°

Green-Blue

90°

Blue

90°

Blue- Violet

90°

Blue

120°

Blue-Violet

120°

Violet

120°

Blue-Violet

150°

Violet

150°

Violet-Red

150°

Violet

180°

Violet-Red

180°

Red

180°

Blue-Violet

210°

Violet

210°

Violet-Red

210°

Blue

240°

Blue- Violet

240°

Violet

240°

Green-Blue

270°

Blue

270°

Blue-Violet

270°

Green

300°

Green-Blue

300°

Blue

300°

Yellow- Green

330°

Green

330°

Green- Blue

330°

Green-Blue Origin

Blue Origin

Blue-Violet Origin

Violet A xis

Violet-Red Axis

Red A xis

Green-Blue

0°

Blue

0°

Blue- Violet

0°

Blue

30°

Blue-Violet

30°

Violet

30°

Blue-Violet

60°

Violet

60°

Violet-Red

60°

Violet

90°

Violet-Red

90°

Red

90°

Violet-Red

120°

Red

120°

Red-Orange

120°

Red

150°

Red-Orange

150°

Orange

150°

Red-Orange

180°

Orange

180°

Orange- Yellow

180°

Red

210°

Red-Orange

210°

Orange

210°

Violet-Red

240°

Red

240°

Red-Orange

240°

Violet

270°

Violet-Red

270°

Red

270°

Blue-Violet

300°

Violet

300°

Violet- Red

300°

Blue

330°

Blue-Violet

330°

Violet

330°

Digitize

Google

Original from
UNIVERSITY OF MICHIGAN

354

TECHNOLOGY OF ART PRODUCTION

180° OF THE SPECTRUM DISTRIBUTED THROUGH 360° (Concluded)

Violet Origin

Red-Orange A xis

Violet

0°

Violet-Red

30°

Red

60°

Red-Orange

90°

Orange

120°

Orange- Yellow

150°

Yellow

180°

Orange- Yellow

210°

Orange

240°

Red-Orange

270°

Red

300°

Violet-Red

330°

Violet-Red Origin

Orange A xis

Violet-Red

0°

Red

30°

Red -Orange

60°

Orange

90°

Orange- Yellow

120°

Yellow

150°

Yellow-Green

180°

Yellow

210°

Orange- Yellow

240°

Orange

270°

Red-Orange

300°

Red

330°

Red Origin

Orange- Yellow A xis

Red 0°

Red-Orange 30°

Orange 60°

Orange-Yellow 90°

Yellow 120°

Yellow-Green 150°

Green 180°

Yellow-Green 210°

Yellow 240°

Orange-Yellow 270°

Orange 300°

Red-Orange 330°

Red -Orange Origin Orange Origin Orange- Yellow Origin

Yellow A xis Yellow-Green A xis Green A xis

Red-Orange

0°

Orange

0°

Orange-Yellow

0°

Orange

30°

Orange- Yellow

30°

Yellow

30°

Orange- Yellow

60°

Yellow

60°

Yellow-Green

60°

Yellow

90°

Yellow-Green

90°

Green

90°

Yellow-Green

120°

Green

120°

Green-Blue

120°

Green

150°

Green- Blue

150°

Blue

150°

Green-Blue

180°

Blue

180°

Blue- Violet

180°

Green

210°

Green-Blue

210°

Blue

210°

Yellow- Green

240°

Green

240°

Green-Blue

240°

Yellow

270°

Yellow-Green

270°

Green

270°

Orange- Yellow

300°

Yellow

300°

Yellow-Green

300°

Orange

330°

Orange- Yellow

330°

Yellow

330°

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

150° OF THE SPECTRUM DISTRIBUTED THROUGH 360°.

Yellow Origin

Green-Green-Blue Axis

Yellow-Green Origin

Grene-Blue-Blue Axis

Yellow

Yellow-Green

Green

Green-Blue

Blue

Blue-Violet
Blue

Green-Blue
Green

Yellow-Green

0°
36°
72°
108°
144°
180°
216°
252°
288°
324°

Yellow-Green
Green
Green-Blue
Blue

Blue- Violet
Violet
Blue- Violet
Blue

Green-Blue
Green

0°
36°
72°
108°
144°
180°
216°
252°
288°
324°

Green Origin

Blue-Blue- Violet A xis

Green 0°

Green-Blue 36°

Blue 72°

Blue-Violet 108°

Violet 144°

Violet-Red 180°

Violet 216°

Blue- Violet 252°

Blue 288°

Green-Blue 324°

Green- Blue Origin

Blue-Violet-Violet Axis

Green-Blue 0°

Blue 36°

Blue- Violet 72°

Violet 108°

Violet-Red 144°

Red 180°

Violet-Red 216°

Violet 252°

Blue-Violet 288°

Blue 324°

Blue Origin

Violet-Violet-Red Axis

Blue

0°

Blue- Violet

36°

Violet

72°

Violet- Red

108°

Red

144°

Red-Orange

180°

Red

216°

Violet-Red

252°

Violet

288°

Blue-Violet

324°

Digitized by

Google

Blue- Violet Origin

Violet- Red- Red Axis

Blue-Violet 0°

Violet 36°

Violet- Red 72°

Red 108°

Red-Orange 144°

Orange 180°

Red-Orange 216°

Red 252°

Violet-Red 288°

Violet 324°
Original from

UNIVERSITY OF MICHIGAN

356

TECHNOLOGY OF ART PRODUCTION

150° OF THE SPECTRUM DISTRIBUTED THROUGH 360° (Concluded)

Violet Origin Violet-Red Origin

Red- Red-Orange Axis Red-Orange-Orange Axis

v loiet

o

v ioiet-i\.ea

U

Violet-Red

36°

Red

36°

Red

72°

Red-Orange

72°

Red-Orange

108°

Orange

108°

Orange

144°

Orange-Yellow

144°

Orange-Yellow

180°

Yellow

180°

Orange

216°

Orange- Yellow

216°

Red-Orange

252°

Orange

252°

Red

288°

Red-Orange

288°

Violet-Red

324°

Red

324°

Red Origin

Orange-Orange- Yellow A xis

Red

0°

Red-Orange

36°

Orange

72°

Orange- Yellow

108°

Yellow

144°

Yellow-Green

180°

Yellow

216°

Orange-Yellow

252°

Orange

288°

Red-Orange

324°

Red -Orange Origin

Orange- Yellow- Yellow Axis

Red-Orange

0°

Orange

36°

Orange- Yellow

72°

Yellow

108°

Yellow-Green

144°

Green

180°

Yellow-Green

216°

Yellow

252°

Orange- Yellow

288°

Orange

324°

Orange Origin

Ytllow- Yellow-Green Axis

Orange

0°

Orange-Yellow

36°

Yellow

72°

Yellow-Green

108°

Green

144°

Green-Blue

180°

Green

215°

Yellow-Green

252°

Yellow

288°

Orange-Yellow

324°

Digitized by^OOglG

Orange- Yellow Origin

Yellow-Green-Green Axis

Orange-Yellow

0°

Yellow

36°

Yellow-Green

72°

Green

108°

Green-Blue

144°

Blue

180°

Green-Blue

216°

Green

252°

Yellow-Green

288°

Yellow

324°

Original from
UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

d. 120° OF THE SPECTRUM DISTRIBUTED THROUGH 360°.

Yellow Origin Yellow-Green Origin ,

Green Axis Green-Blue Axis

Yellow

0°

Yellow-Green

0°

Yellow-Green

45°

Green

45°

Green

90°

Green-Blue

90°

Green-Blue

135°

Blue

135°

Blue

180°

Blue-Violet

180°

Green-Blue

225°

Blue

225°

Green

270°

Green-Blue

270°

Yellow-Green

315°

Green

315°

Green Origin

Green-Blue Origin

Blue Axis

Blue-Violet Axis

Green

0°

Green-Blue

0°

Green-Blue

45°

Blue

45°

Blue

90°

Blue-Violet

90°

Blue- Violet

135°

Violet

135°

Violet

180°

Violet-Red

180°

Blue-Violet

225°

Violet

225°

Blue

270°

Blue-Violet

270°

Green-Blue

315°

Blue

315°

Blue Origin

Blue-Violet Origin

Violet Axis

Violet- Red Axis

Blue

0°

Blue- Violet

0°

Blue-Violet

45°

Violet

45°

Violet

90°

Violet-Red

90°

Violet-Red

135°

Red

135°

Red

180°

Red-Orange

180°

Violet-Red

225°

Red

225°

Violet

270°

Violet-Red

270°

Blue-Violet

315°

Violet

315°

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

358

TECHNOLOGY OF ART PRODUCTION

120° OF THE SPECTRUM DISTRIBUTED THROUGH 360° (Concluded)

Violet Origin Violet-Red Origin

Red Axis

Red-Orange A

xis

Violet

0°

Violet-Red

0°

Violet-Red

45°

Red

45°

Red

90°

Red-Orange

90°

Red-Orange

135°

Orange

135°

Orange

180°

Orange- Yellow

180°

Red -Orange

225°

Orange

225°

Red

270°

Red-Orange

270°

Violet-Red

315°

Red

315°

Red Origin

Orange Axis

Red

0°

Red-Orange

45°

Orange

90°

Orange- Yellow

135°

Yellow

180°

Orange-Yellow

225°

Orange

270°

Red -Orange

315°

Orange Origin

Yellow Axis

Orange

0°

Orange-Yellow

45°

Yellow

90°

Yellow-Green

135°

Green 180°

Yellow-Green 225°

Yellow 270°

Orange- Yellow 315°

Red-Orange Origin

Orange- Yellow A xis

Red-Orange 0°

Orange 45°
Orange- Yellow 90°

Yellow 135°

Yellow-Green 180°

Yellow 225°

Orange-Yellow 270°

Orange 315°

Orange- Yellow Origin

Yellow-Green Axis

Orange- Yellow 0°

Yellow 45°

Yellow-Green 90°

Green 135°

Green-Blue 180°

Green 225°

Yellow-Green 270°

Yellow 315°

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

SELECTIVE SYSTEMS

90° OF THE SPECTRUM DISTRIBUTED THROUGH 360°.

Yellow Origin Yellow-Green Origin

Yellow-Green-Green Axis Green-Green-Blue Axis

Yellow

Yellow-Green

Green

Green-Blue

Green

Yellow-Green

0°
60°
120°
180°
240°
300°

Yellow-Green
Green
Green-Blue
Blue

Green-Blue
Green

0°
60°
120°
180*
240°
300°

Green Origin

Green-Blue-Blue Axis

Green 0°

Green-Blue 60°

Blue 120°

Blue- Violet 180°

Blue 240°

Green-Blue 300°

Green-Blue Origin

Blue-Blue-Violet Axis

Green-Blue 0°

Blue 60°

Blue-Violet 120°

Violet 180°

Blue-Violet 240°

Blue 300°

Blue Origin

Blue- Violet- Violet A xis

Blue 0°

Blue-Violet 60°

Violet 120°

Violet-Red 180°

Violet 240°

Blue-Violet 300°

Blue-Violet Origin

Violet-Violet- Red Axis

Blue-Violet 0°

Violet 60°

Violet-Red 120°

Red 180°

Violet-Red 240°

Violet 300°

Violet Origin

Violet- Red- Red Axis

Violet 0°

Violet-Red 60°

Red 120°

Red-Orange 180°

Red 240°

Violet-Red 300°

Violet-Red Origin

Red- Red-Orange Axis

Violet-Red 0°

Red 60°

Red-Orange 120°

Orange 180°

Red-Orange 240°

Red 300°

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90° OF THE SPECTRUM DISTRIBUTED THROUGH 360° (Concluded)

Red Origin

Red-Orange Origin

Red-Orange-Orange A

xis

Orange-Orange- Yellow Axis

Red

0°

Red-Orange 0°

Red-Orange

60°

Orange 60°

Orange

120°

Orange-Yellow 120°

Orange- Yellow

180°

Yellow 180°

Orange

240°

Orange-Yellow 240

Red-Orange

300°

Orange 300°

Orange Origin

Orange- Yellow Origin

Orange- Yellow- Yellow A xis

Yellow- Yellow-Green Axis

Orange

0°

Orange-Yellow 0°

Orange- Yellow

60°

Yellow 60°

Yellow

120°

Yellow-Green 120°

Yellow-Green

180°

Green 180°

Yellow

240°

Yellow-Green 240°

Orange- Yellow

300°

Yellow 300°

f. 60° OF THE SPECTRUM DISTRIBUTED THROUGH 360 c

Yellow Origin

Yellow-Green Axis

Yellow

Yellow-Green

Green

Yellow-Green

0°
90°
180°
270°

Yellow-Green Origin

Green Axis

Yellow-Green
Green
Green-Blue
Green

0°
90°
180°
270°

Green Origin

Green-Blue Axis

Green

Green-Blue

Blue

Green-Blue

0°
90°
180°
270°

Green-Blue Origin

Blue Axis

Green-Blue
Blue

Blue-Violet
Blue

0°
90°
180°
270°

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OF THE SPECTRUM DISTRIBUTED THROUGH 360° (Concluded)

Blue Origin

Blue- Violet Axis

Blue

Blue-Violet,

Violet

Blue-Violet

0°
90°
180°
270°

Blue-Violet Origin

Violet Axis

Blue-Violet
Violet

Violet-Red
Violet

0°
90°
180°
270°

Violet Origin

Violet- Red Axis

Violet

Violet-Red

Red

Violet-Red

0°
90°
180°
270°

Violet-Red Origin

Red Axis

Violet-Red
Red

Red -Orange
Red

0°
90°
180°
270°

Red Origin

Red-Orange Axis

Red 0°

Red-Orange 90°

Orange 180°

Red-Orange 270°

Orange Origin

Orange- Yellow Axis

Orange 0°

Orange- Yellow 90°

Yellow 180°

Orange- Yellow 270°

Red-Orange Origin

Orange Axis

Red-Orange 0°

Orange 90°

Orange-Yellow 180°

Orange 270°

Orange- Yellow Origin

Yellow A xis

Orange-Yellow 0°

Yellow 90°

Yellow-Green 180°

Yellow 270°

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TECHNOLOGY OF ART PRODUCTION

g. 30° OF THE SPECTRUM DISTRIBUTED THROUGH 360°.
Yellow Origin Yellow-Green Origin

Yellow- Yellow-Green Axis

Yellow
Yellow-Green

0°
180°

Yellow-Green-Green Axis

Yellow-Green
Green

0°
180°

Green Origin

Green-Green-Blue Axis

Green 0°
Green-Blue 180°

Green-Blue Origin

Green-Blue-Blue Axis

Green-Blue 0°
Blue 180°

Blue Origin

Blue-Blue-Violet Axis

Blue 0°
Blue- Violet 180°

Blue-Violet Origin

Blue- Violet- Violet A xis

Blue-Violet 0°
Violet 180°

Violet Origin

Violet-Violet- Red Axis

Violet 0°
Violet-Red 180°

Violet-Red Origin

Violet- Red- Red Axis

Violet-Red 0°
Red 180°

Red Origin

Red- Red-Orange Axis

Red 0°
Red-Orange 180°

Red-Orange Origin

Red-Orange-Orange Axis

Red-Orange 0°
Orange 180°

Orange Origin

Orange-Orange- Yellow Axis

Orange 0°
Orange- Yellow 180°

Orange- Yellow Origin

Orange- Yellow- Yellow Axis

Orange- Yellow 0°
Yellow 180°

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CHAPTER 2

PRODUCTION OF DESIGN

A. Elements of Linear Design

GEOM ETRIC point has no physical extension . An artistic point is the physi-

*■ cal expression of a geometric point in a given material medium. This medium
renders it visible. We can assume this point to be xr 2 , where r approaches zero.
Different media require different dimensions for r. A scale moving toward zero
may be evolved from a wide flat brush to a pointed one, to a pastel edge, and
finally, to the finest pencil point.

A point moving uniformly on a plane produces a visible trajectory. This
trajectory is a linear design. The time required to evolve such a trajectory
and the speed of movement determine its linear dimensions.

A point can move only through an angle. When the angle equals zero de-
grees, the point does not move. When the angle equals 180°, the point moves
indefinitely and the resulting trajectory is an unending straight line. When
the time of the movement of a point under 180° is limited, the resulting
trajectory is a rectilinear segment of a definite extension. This extension can
be measured in terms of linear measurement. Thus, we obtain the first element
of a linear design: rectilinear segment of a definite dimension (xi).

Continuous progression of rectilinear segments moving in one direction pro-
duces closed forms, or with a tendency to close, according to the arithmetical
property of the angles under which they move, each angle being the divisor
of a dividend of 180°, or its multiple, with various coefficients.

Continuous progression of rectilinear segments moving in alternating direc-
tions produces broken lines when the segments change their direction under one
constant angle. Their range of extension is from the dimension of a selected
segment unit (Z0°) to infinity (ZlS0°), and back to the dimensions of the
unit (Z360°).

A linear element moving through an angle extends itself through the second
coordinate (xj) 1 , thus evolving on a plane. Angle and direction become the two
other elements of linear design. Dimension and angle allow an infinite number
of variations. Direction can only be clockwise or counter-clockwise.

A design evolving through angles moves in the direction opposite to the
geometric formation of the angles.

' x t = width.

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TECHNOLOGY OF ART PRODUCTION

Figure 1. Design evolving through angles.

When angles form in a clockwise direction (O). the design moves counter-
clockwise (O). and vice- versa.

Curvilinear configurations are the derivatives of rectilinear configurations,
each arc being produced from a rectilinear segment as a radius or a diameter.
Dimensions of rectilinear segments determine the curvature of the corresponding
arcs. Angular values determine the dimensions of the corresponding arcs. The
increase of a radius decreases the curvature of a corresponding arc. The increase
of an angle increases the dimension of a corresponding arc.

4 10' X-3 , A 10*

Figure 2. Increase of angle increases dimension of corresponding arc.

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B. Rhythmic Design

The human idea of the universe changes through the ages. There have
been radical changes from Ptolemy to Eddington. The dominant tendency in
modern science is to "re-create" the universe geometrically, i.e., to explain
all physical phenomena as derivatives of the properties of space. The idea of
the homogeneity of matter and space was introduced first by Democritus in
philosophical rather than physical form. Many centuries elapsed until Descartes
established this postulate on a physical basis. Nevertheless, Newton challenged
it. The mathematical equipment of Descartes' epoch, 39 years before Leibnitz
invented differential calculus (1676), was insufficient for the mathematical inter-
pretation of the postulate. Recent progress in science, due to the contributions
of Riemann, Minkowsky, Lorentz, Einstein, Michelson, Millikan, Compton and
Eddington, makes possible a universe of highly complex space. It is known as a
space-time continuum, where time is one of the components of space.

Any art-form is also a derivative of the space-time continuum. Art can be
measured and analyzed like any other phenomenon of our universe. Any
analytical result can be represented graphically, i.e., geometrically, in short,
in terms of space-time relations. The esthetic value, from the analytical point
of view, is a function of the space-time relations. Design is geometrically the
most obvious art-form, since the idea and the realization in an art medium are
both accomplished in empirical space.

A design is rhythmic if analysis reveals the regularity in the sequence of its
components and their correlations. The regularity or the irregularity of sequence
may be obtained in any finite continuity. The simplest form of regularity can
be determined by a constant relation between two consecutive terms. When
this relation is unity, we have the simplest case of regularity. In wave theory
it is known as "simple harmonic motion"; it is a uniform motion in which the
consecutive terms are related as one to one. It may be determined as monomial
periodicity and expressed as a+a-fa . . . More developed forms of rhythm
may be observed in a continuity where regularity can be deduced from relations
of the groups of terms, for example (a+b-f c) -f (a+b-f c) ... In this case,
in order to discover the form of regularity, it is necessary to observe at least
six terms. A non-rhythmic continuity may be determined as a sequence where
regularity can be observed only in an infinite number of terms. Irregularities
producing regular sequences should be considered as rhythmic deviations and
variations.

The general method of producing rhythmic sequences is based on the physical
phenomenon known as interference. Two harmonic waves of the same period
(frequency) acting simultaneously do not produce any new sequence. In order
to obtain new sequences from two or more waves, it is necessary that their
periods should be different, even when they are harmonic. This requirement
will be fulfilled when the two or more periodicities are not related as 1 : 1 : 1 : . . . .

A wave resulting from an interference of any periodicities, related as a : 1,
where any value may be attributed to a, does not produce a new sequence of
values but only of intensities. In order to obtain new sequences of values as

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TECHNOLOGY OF ART PRODUCTION

well as of intensities, unity must be excluded from the previous relation. It
will assume the form A = B, or A = B = C, etc., where none of the terms
equals unity. These rhythmic sequences of values and intensities produce series,
which result from interference of the two or more component periodicities. The
procedure of obtaining the resulting rhythmic series is based on the synchroniza-
tion of the component periodicities by means of their common denominator or
common product. Taking A and B as the two component periodicities, we obtain
the product AB which represents the total of units on which the synchronization
can be performed. A is taken B times against B taken A times, i.e., if A «= 3
and B = 2, then AB = 6. Thus 3 will follow 2 times against 2 following 3 times.

~* i l_t

r~i_i

3 * ^

Figure 3. Interference of periodicities .

2x3

If we draw the result by dropping perpendiculars to both of the two com-
ponent periodicities through their consecutive terms, we obtain the following
series: 2 + 1 + 1+2.

1_J L

Digiti

G

Figure 4. Resulting periodicity.

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The interference of 4 against 3, obtained by the method described above,
will produce 3 + 1 + 2 + 2 + 1+3.

1

i

1 1

! i

i i
i i

i

i i

J L

Figure 5. Resulting periodicity of 4 + 3.

The elements of linear design are angles, dimensions, directions and their
derivatives (arcs, sectors, etc.). under which a point realized in an art medium
moves through an area.

The rhythmic series obtained above, being applied to dimensions of the
rectilinear segments moving in a constant direction under a 90° angle, will
produce the following results:

4:3

3 12.

Figure 6. 3 + 2 applied to rectilinear
segments moving in a constant di-
rection under 90° angle.

Digit

oogle

Figure 7. 4 + 3 applied to rectilinear
segments moving in a constant di-
rection under 90° angle.

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TECHNOLOGY OF ART PRODUCTION

Applying the same series to the progression of angles, where the angle unit
equals 30°, we obtain the following results.

The sequence of angles follows the linear periodicity
60° (2) -I- 30° (1) + 30° (1) + 60° (2)

3 1*

Figure 8. 3 + 2 applied to rectilinear segments
moving under angle unit of 30°.

The selection of angles accords with the sequence of periodicities:
90° (3) + 30° (1) + 60° (2) + 60° (2) + 30° (1) + 90° (3)

o
1

60*

4:3

>

60°

.50"

Digiti

Figure 9. 4 + 3 applied to rectilinear segments

unit of 30°.

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PRODUCTION OF DESIGN

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The simple process of evolving circular arcs of 180°, using the previous
schemes as diameters and moving in one direction, will result in the following
design.

4:3.

Figure 10. 3 + 2 and 4 + 3 moving in one direction with circular arcs of 180°.

Different variations of the linear designs, once obtained, may be achieved
by means of the displacement of the terms in a rhythmic series. Thus,

2 + 1 + 1+2 (see Figure 4) will produce the following displacements:
l + l+ 2+ 2;l+2 + 2 + l;2 + 2+ l + l; while the series 3 + 1 + 2 +
+ 2 + 1+3 (see Figure 5) will produce these: 2 + 2 + l+ 3 + 3 + l;l +
+ 2 + 2 + 1+ 3 + 3; 2 + 1+ 3 + 3 + 1+ 2; 1+3 + 3 + 1+ 2 + 2;

3 + 3 + 1+ 2 + 2 + 1. These mathematical variations, applied to dimensions
or angles, will produce corresponding variations in design.

A more complex form of variation may be obtained by the distributive use
of algebraic powers. For instance, (2 + 1 + 1 + 2)* = (4 + 2 + 2 + 4) +
+ (2 + 1 + 1 + 2) + (2 + 1 + 1 + 2) + (4 + 2 + 2 + 4). Applying these
new periodic series to the components of linear design, we obtain a multiple
motive, which harmoniously repeats the relations observed in the primary one.

All these methods may be applied to any linear design evolved on an area
without boundaries. In a given area, the relation of its sides, or coordinates,
will determine the behavior of linear and angular values, and result in a rhythmic
design conditioned by the properties of the area. For example, in a square
area, moving clockwise through the 4 : 3 series of angles, we obtain a different
result from that of an oblong, where the relation of the two sides is 2 : 1 .

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370 TECHNOLOGY OF ART PRODUCTION

Figure 11. Square and oblong areas contrasted.

& : / AREA

Figure 12. 2:1 area.

The effect produced by the powers applied to the sum of the sides of an
oblong is to split each of the two sides into partial segments, which are in the
same relation to each other as the sides of the oblong. Connecting these partial
segments under a right angle, we produce rectangular coordinates. The point of
intersection of these is the rhythmic center of the area.

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(1 IV 4

2 2 1
+ 9 + 9 + 9

2

Figure 13. Powers applied to the sum of the sides of an oblong.

The rhythmic center determines the origin and the behavior of the linear
design obtained through angular relation. This process may be indefinitely con-
tinued by application of the higher powers. There will be four rhythmic centers
produced for the third power, sixteen for the fourth, etc.

<

)

Figure 14. Rhythmic center of a rectangle.

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TECHNOLOGY OF ART PRODUCTION

There are many other processes in evolving a rhythmic design through its
elements, or through the properties of its area. There are some continuous pro-
cedures based on the recurrence of one process, such as:

(1) Dropping perpendiculars from the vertices of the two triangles produced
by the diagonal of an oblong.

Figure 15. Evolving a rhythmic design on the basis of the diagonal

of a rectangle.

(2) Producing continuous sequence of arcs where the radius may be selected
from any segment, connecting the rhythmic center with one of the sides of an
oblong. The consecutive origins are formed by the tangent points of arcs touching
the sides of an oblong.

Figure 16. Continuous sequence of arcs.

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Geometrical variation of a given design may be obtained by means of :

(1) Consecutive displacement of the partial power areas;

(2) revolving an area around one of the sides;

(3) revolving areas around the rhythmic centers;

(4) displacement in the direction of the coordinates and the rays connecting
the sides with the rhythmic center.

Various other forms of areas, such as the circle, the ellipse or any irregular
area, may be determined in their proportions by its axes. Any design may be
inscribed or described through these axes. Various deviations from symmetry
may be obtained by the translation of a motive from an area of one type of
structure to that of another.

/ :/ 2 :/

Figure 17. Translating a motive from an area of one type
of structure to another.

The same method of rhythmic series is applicable to color in its components
(hue, value, intensity). In order to treat one of these color components rhyth-
mically, it is necessary to select a scale within the given limits. This spectral
(or value) scale may be inscribed in a circle. The center of the color-wheel will
designate the origins of the corresponding angles which are the foundation of
a design to which color is to be applied. The rhythmic progression on the twelve
step spectral scale may be expressed through the values of the arcs of the circular
color chart. For example, assuming a circular scale in the following sequence:

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TECHNOLOGY OF ART PRODUCTION

yellow, yellow-green, green, blue-green, blue, blue-violet, violet, red-violet,
red, red-orange, orange, yellow-orange,

and assuming yellow to be the initial color (t. e., 0°), the rhythmic series
3 + 1+ 2 + 2 + 1+ 3 will equal 90° + 30° + 60° + 60° + 30° + 90°. This
will result in the following series of hues:

Yellow, blue-green, blue- violet, red, red-orange, yellow.

Figure 18. Twelve step spectral scale.

In this way, correlation between the rhythm of design and that of color
may be achieved by the identical method of rhythmic sequence of the elements
of space.

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C. Design Analysis

375

/. Fish Panel
5-5-3 Rhythmic centers — 3rd power.

1.

5-5-2 Periodicity — clockwise — unit of angle 20°; unit of length of lines 2" —
arbitrary construction of circles and a square — the periodicity of 5-4-2 drawn
twice within the square form. Arbitrary choice of color.

2.

5-5-2 Periodicity — clockwise — unit of angle 20°; unit of lines 2" — arbitrary
construction of circles and squares, retaining all lines of periodicity. Color
arbitrary choice.

3.

5-5-4 Periodicity — counterclockwise — unit 1/2"; unit of angle 10° — con-
struction of equilateral triangles on each line. Color — three sets of triads —
changing in value.

4.

5-5-3 Periodicity — clockwise — unit of angle 15°; unit of line 1" — arbitrary
construction of square, triangle and circles, retaining some lines of the periodicity.
Arbitrary choice of color.

5.

Alternating }f. S — 50° and 30° clockwise — length of sides based on 5 -5- 7 perio-
dicity with 1" as unit. Squares divided into stripes 5-5-3. Color scheme — center
unit of 5 -5- 3 periodicity green and moving from green in both directions — blue,
violet, red, orange and yellow.

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Z. 4+3 Rectangle

Construction: Rhythmic center determined first. Using periodicity 4-5-3
(3 + 1+ 2+ 2 + 1+3) and 20° as a unit; angles drawn clockwise in each
quadrant, using the coordinate (from the rhythmic center to the side of the
rectangle) as first side of angle of 60°. In entire rectangle, same angular scheme
was developed in each quadrant.

Color: For application of color, each quadrant was developed individually.
An angle of 15° used as a unit, and six such angles drawn from rhythmic center to
outer boundaries of each quadrant. Each area painted in a different set of two
colors. Each set of colors chosen from the color wheel, from points moving
clockwise at right angles.

Value : Divisions for values drawn clockwise in each quadrant and parallel
to the coordinate of the quadrant. The spacing of value areas determined by the
application of the periodicity of 4-i-3. Six units to the periodicity, and six values
in each quadrant. The lowest value placed nearest the coordinate in each quad-
rant.

( See preceding page )

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3. 3+2 Rectangle

Construction: Rhythmic center first determined. Sides of each quadrant
divided into 24 units. Then the crossing lines drawn. In the square quadrant,
the design is drawn normally. In each of the other 3 quadrants, the design is
redrawn in distorted position, as dictated by different spacing.

Color: Four colors in each quadrant. Color wheel with sixteen colors
developed.

Omitting orange yellow and blue violet, a group of four colors for each quadrant
chosen from the four points of 2 diameters that cross at one unit less than a
right angle (as shown above). Y, BG, V and OR used in first quadrant. These
four colors arranged so that the one of lowest natural value would be placed in
the same area in each quadrant. This applied to the placement of all colors.

Value: The value areas are equal; but in each quadrant, there are a differ-
ent number, according to the proportions of the sides of the 3-f-2 rectangle,
(9 -s-6 -T- 6-^4). These areas are arranged perpendicular to the coordinate of each
quadrant and move around the rectangle in clockwise direction.

(Sec preceding page)

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4. 3+2 Rectangle

Construction: Division into 576 proportionate areas. Rhythmic center
determined. Division of each quadrant into 576 proportionate areas. In square
quadrant landscape is drawn, and repeated in other three quadrants and in
large rectangle in the first power.

Color: Chose four sets of colors for 4 quadrants.

yo - Y - yg

o

g

R

GB

r

b

| rv - V - bv

Yellow represents Spring
GB '* Summer

V " Winter

RO " Fall

Rendering of landscape in first power in white represents Snow. Colors dis-
tributed in same succession, a, b, and c, always occurring in same areas. The color
to left of dominant color always occurring at — a; the dominant color — b; and
color to right of dominant color — c.

Values: There are two values of a and two values of c.

(See preceding page)

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TECHNOLOGY OF ART PRODUCTION

5. Instruments

3-f-2 rectangle. Rhythmic center determined. Violin drawn normally in
square quadrant, divided into 576 spaces. Same subject redrawn in other 3
quadrants in distortion determined by network of spaces. Large instrument
drawn on coordinates of rhythmic center with normal section in square area.
Bows drawn on diagonal of each quadrant, stemming from the rhythmic center.

12 divisions in each quadrant for values.

15 divisions in each quadrant for intensity.

For background, Red Orange — dominant color, modulated by its comple-
ment — Blue Green. For instruments, mixture of Red Orange with small amount
of Blue Green of normal value for center of instrument, changing intensity and
values as the edges are approached. Ornaments on instrument: — mixture of Red
Orange with small amount of Blue Green in a medium value, changing value as
the edges are approached.

(See preceding page)

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Figure 24, Scale of saturation produced by linear configurations in a confined area.

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TECHNOLOGY OF ART PRODUCTION

PRODUCTION OF DESIGN

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PRODUCTION OF DESIGN

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!

4
1

3c

■a

rv -■ _j l f^rinrslf* Original from

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391

D. Three Compositions in Linear Design
One origin point for two subjects.
Family: ■§■ series.

Clockwise progression for both subjects. Variation of direction occurs with
the recurrence of values. All number-values express radii and the length of arcs.
Radii and arcs are in direct relation.

First Subject:

The first power trinomial synchronized with its distributive square. The
group: 6=4 + 1+1.

Values: 6(4 + 1 + 1) = 24 + 6 + 6

6(1 + 4 + 1) = 6 + 24 + 6

6(1+1+4) =6 + 6 + 24

Second Subject :

The distributive square of the same group (4 + 1 + 1).

Values: (4 + 1 + l) 2 = (16 + 4 + 4) + (4 + 1 + 1) + (4 + 1 + 1)
(1 + 4 + \y = (1 + 4 + 1) + (4 + 16 + 4) + (1 + 4 + 1)
(1 + 1 + 4) 2 = (1 + 1 + 4) + (1 + 1 + 4) + (4 + 4 + 16)

Linear unit for the radii equals 1/4".

Arc unit = 10°.

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TECHNOLOGY OF ART PRODUCTION

Figure 29. Linear design 6/6 series.

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One origin point for all groups: the center of a diameter whose length is 12".
Family: series.

Clockwise progression for all groups. Variation of direction in each linear
group takes place with each term. All number-values express radii and the length
of arcs. Radii and arcs are in direct relation.

The following number-values represent binary, ternary and quinternary groups
in the order of their appearance. The degrees indicate the origin-axes for each
group in relation to the ordinate, which is zero-axis.

Binomials:

7+5

0°\

5+7

180°]"

5+2+5

o°l

2+5+5

120° }

5+5+2

240°J

2+3+2+3+2

0°"

2+2+3+2+3

72°

3+2+3+2+2

144° -

2+3+2+2+3

216°

3+2+2+3+2

288°

J

Linear unit = 1/2".
Arc unit = 10°.

two sectors in symmetry

three sectors in symmetry

five sectors in symmetry

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UNIVERSITY OF MICHIGAN

PRODUCTION OF DESIGN

395

One origin point for all groups.
Family : series.

Clockwise progression for all groups. Variation of direction occurs with the
recurrence of values. All number-values express radii and the length of arcs. Radii
and arcs are in direct relation.

Groups in the order of their appearance:

19 + 17
17 + 19

17+2 + 17
2 + 17 + 17
17 + 17+2

2 + 15 +2 + 15+2
15+2 + 15+2+2
* 2 + 15+2 + 2 + 15
15+2+2+15 + 2
2+2 + 15+2 + 15

2 + 2 + 11+2+2 + 2 + 11+2+2
2 + 11+2+2+2 + 11+2+2+2
11+2+2+2 + 11+2+2+2+2
2+2+2 + 11+2+2+2+2 + 11
2 + 2 + 11+2+2+2+2 + 11+2
2 + 11+2+2+2+2 + 11+2+2
11+2+2+2+2 + 11+2+2+2
2+2+2 + 2 + 11+2+2+2 + 11
2+2 + 2 + 11+2+2 + 2 + 11+2

Linear unit = 1/4".

Arc unit = 10°.

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

396

TECHNOLOGY OF ART PRODUCTION

PRODUCTION OF DESIGN

397

E. Problems in Design

(1) Rhythmic series of angles. Overlapping progressions of striped areas:
stripes parallel or perpendicular to one of the sides. Color progressions on
same areas.

(2) Rhythmic series of angles used :

(a) as radii

(b) as diameters.

(3) The cross section segments used as radii or as diameters, combined with the
striping or coloring.

Rhythmic progression of values only (black — white) or combined with
hues.

(4) Using halves of rhythmic series of angles in their crossings as determining
boundaries for a motive.

(5) Developing the same procedure with the second power of rhythmic series
of angles.

(6) Evolving any motive in the second power areas.

(7) Evolving a motive in the first and the second power areas.

(8) Clockwise or counter-clockwise revolving of phe four motives in the second
power areas.

(9) Various combinations through the four positions in the second power areas

a

b

a

b

d

c

d

c

a

b

a

b

d

c

d

c

(10) Using a motive twice: once — in the whole area, and once — in one of the
second power areas.

Using a motive three times: once in the whole area, and twice in different
second power areas.

Using a motive four times: once in the whole area, and three times in the
second power areas.

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TECHNOLOGY OF ART PRODUCTION

(11) Bithematic Composition

One theme in the area of the first power, one theme (repeated through
quadrants) in the second power. Various methods of overlapping the two
themes.

(12) Rhythmic series of angles used in the second power through the whole area
and in the first power through the second power areas.

(13) Rhythmic distortions of the subject in various second power areas (with
revolving, and without).

(14) Consecutive displacement of the subject where rhythm represents the
number of the cross section units in the second power areas. Clockwise and
counter-clockwise displacement.

(15) Distortion of a subject of the first power through the proportions of the
second power areas.

(16) All of the above forms applied to the third power. Bithematic and tri-
thematic composition, using the first, the second and the third power.

Original from
UNIVERSITY OF MICHIGAN

CHAPTER 3

PRODUCTION OF MUSIC

A. Coordination of Temporal Structures 1

"X ^"OTION — that is, changeability in time — is the most important intrinsic
property of music. Different cultures of different geographical and his-
torical localities have developed many types and forms of intonation. The latter
varies greatly in tuning, in quantity of pitches employed, in quantity of simul-
taneous parts, and in the ways of treating them.

The types are as diversified as drum-beats, instrumental and vocal monody
(one part music), organum, discantus, counterpoint, harmony, combinations of
melody and harmony, combinations of counterpoint and harmony, different
forms of coupled voices, simultaneous combinations of several harmonies, and
many others. Any of these types — as well as any combinations of them —
constitute the different musical cultures. In each case, musical culture crystal-
lizes itself into a definite combination of types and forms of intonation. The
latter crystallize into habits and traditions.

For example, people belonging to a harmonic musical culture want every
melody harmonized. But people belonging to a monodic musical culture are
disturbed by the very presence of harmony. Music of one culture may be music
(meaningful sound) to the members of that culture; but the very same music
may be noise (meaningless sound) to the members of another. The function-
ality of music is comparable to a great extent to that of a language.

Nevertheless, all forms of music have one fundamental property in common:
organized time. The plasticity of the temporal structure of music, as expressed
through its attacks and durations, defines the quality of music. Different types
and forms of intonation — as well as different types of musical instruments —
come and go like the fashions, while the everlasting strife for temporal plasticity
remains a symbol of the "eternal" in music.

'This chapter concerns one basic aspect of Music by Means of Geometrical Projection,

musical composition, the temporal structure of Book 4: Theory of Melody. Book 5: Special

music as it pertains both to simultaneity and Theory of Harmony. Book 6: The Correlation

continuity. The problem is one of coordinating of Melody and Harmony. Book 7: Theory of

the rhythmic elements of individual parts with Counterpoint. Book 8: Instrumental Forms,

the structure of the score as a whole. For com- Book 9: General Theory of Harmony. Book 10:

prehensive analysis of every phase of the art of Evolution of Pitch-Families (Style). Book 11:

composition, the reader is referred to The Schil- Theory of Composition. Book 12: Theory of

linger System of Musical Composition, published Orchestration. (Ed.)

in two volumes by Carl Fischer, Inc. The scope *Fron Book 1, "Theory of Rhythm," of The
of this work is suggested by the titles of its twelve SchiUinger System of Musical Composition. Copy-
branches. Book 1: Theory of Rhythm. Book 2: right 1941 by Carl Fischer, Inc. Reprinted by
Theory of Pitch Scales. Book 3: Variation of permission.

f rtj^inlf? Original from

jUUgH399 UNIVERSITY OF MICHIGAN

I

400 TECHNOLOGY OF ART PRODUCTION

The temporal structure of music, usually known as rhythm, pertains to two
directions: simultaneity and continuity. The rhythm of simultaneity is a form of
coordination among the different components (parts). The rhythm of continuity
is a form of coordination of the successive moments of one component (part).

People of our civilization have developed the power of reasoning at the
price of losing many of the instincts of primitive man. Europeans have never
possessed the "instinct of rhythm" with which the Africans are endowed. So-
called European "classical music" has never attained the ideal it strived for,
that ideal being: the utmost plasticity of the temporal organization. When
J. S. Bach, for example, tried to develop a coordinated independence of simul-
taneous parts, he succeeded in producing only a resultant which is uniformity. 1
We find evidence of the same failures in Mozart and Beethoven. But a score
in which the several coordinated parts produce a resultant which has a distinct
pattern — has been a "lost art" of the aboriginal African drummers. The age of
this art can probably be counted in tens of thousands of years!

Today in the United States, owing to the transplantation of Africans to this
continent, there is a renaissance of rhythm. Habits form quickly — and the
instinct of rhythm in the present American generation surpasses anything
known throughout European history. Yet our professional "coordinators of
rhythm," specifically in the field of dance music, are slaves to, rather than
masters of, rhythm. There is plenty of evidence that the urge for coordination
of the whole through individualized parts is growing. The so-called "pyramids"
(sustained arpeggio produced by successive entrances of several instruments)
is but an incompetent attempt to solve the same problem.

Fortunately, we do not have to feel discouraged or moan over this "lost
art." The power of reasoning offers us a complete scientific solution.

This problem can be formulated as the distribution of a duration-group
through instrumental and attack-groups.

The entire technique consists of five successive operations" with respect to
the following:

(1) The number of individual parts in a score;

(2) The quantity of attacks appearing with each individual part in suc-
cession ;

(3) The rhythmic patterns for each individual part;

(4) The coordination of all parts (which become the resultants of instru-
mental interference) into a form which, in turn, results in a specified
rhythmic pattern (the resultant of interference of all parts); and

(5) The application of such scores to any type of musical measures (bars).

Any part of such a score can be treated as melody, coupled melody, block-
harmony, harmony, instrumental figuration — or as a purely percussive (drum)

'That is to say, when the separate rhythms
of the separate parts of a Bach score are "added
up." the result tends to be simple uniformity.
Schillinger suggests the desirability of scores, and

Digitized byGoOgle

develops a method of scoring, so that the separate
parts, while satisfactory rhythmically by them-
selves, all "add up" to a new rhythm which is
not uniformity. (Ed.)

Original from
UNIVERSITY OF MICHIGAN

PRODUCTION OF MUSIC

401

part. Aside from the temporal structure of the score, the practical uses of this
technique in intonation depend on the composer's skill in the respective fields
concerned, i.e., melody, harmony, counterpoint and orchestration.

B. Distribution of a Duration-Group (T) through Instrumental (i)
and Attack (a) groups

Notation

pli number of places in the instrumental group,
pla number of places in the attack-group.
a a number of attacks in the attack-group.
a T number of attacks in the duration-group.
PL the final number of places.

A the synchronized attack-group (the number of attacks synchronized with
the number of places).

A 1 the final attack group (number of attacks synchronized with the duration-
group).

T the original duration-group.

T' the synchronized duration-group.

T" the final duration-group.

N T " the number of final duration-groups.

Procedures:

(1) Interference between the number of places in the instrumental group (pli)
and the number of places in the attack-group (pla).

pL = pH pla (pli)
pla' pli (pla)

(2) The product of the number of attacks in the attack group (a a ) by the
complementary factor to the number of places in the attack-group (pli
after reduction).

A = a a -pli

(3) Interference between the synchronized attack-group (A) and the number
of attacks in the original duration-group (a T ).

A , = A = a a pli
a T a T

(4) The product of the original duration -group (T) by the complementary
factor to its number of attacks (A').

T = t .a' .I^e5

Digitized byGoOgle

a T

Original from

UNIVERSITY OF MICHIGAN

402

TECHNOLOGY OF ART PRODUCTION

(5) Interference between the synchronized duration-group (T') and the final
duration-group (T").

T

N T" = Y»

C. Synchronization of an Attack-Group (a) with a Duration-Group (T)

Distribution of attacks of an attack-group (a a ) through the number of attacks
of a duration-group (a T ).

First Case: a a
a-j-

A = a x
T'= T

Example:

a, = 4a; T = r 3+2 = 6t; a T = 4a

A = 4a
T' = 6t

h. { i T= r err | \ 1 J n

Figure 1. Synchronization of an attack-group with a duration-group.

Second Case: a.

— ^ 1
a T

A = a T -a a
T'= Ta a

a, = 5a; T = r 3+2 = 6t; a T = 4a

f A = 5a-4 = 20a

T' = 6t-5 = 30t

ns -■ _j l r\r\cs\c> Original from

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PRODUCTION OF MUSIC

403

a

T:fcrr

r-0

Figure 2. Another illustration of synchronization.

Third Case: a a a a < i.e., a reducible fraction
a*T af'

A = a T -a a -

T' = T-a a -

a, = 6a; T = r 3+2 = 6t; a T = 4a

A = 4a -3 = 12a

T = 6t-3 = 18t

•a:

1m, h'i »'

Figure 3. A third illustration of synchronization.

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

404 TECHNOLOGY OF ART PRODUCTION

D. Distribution of a Synchronized Duration-Group (T') through the
Final Duration-Group (T")

First Case: T'

T" = T'

Example:

T = 6t; T" - 6t
6t = 6t

fj p gJ ii iirflJ ii iir^ a

Figure 4. Duration-group distribution.

Third Case: T_
T*

Digitized byGoOgle

T' i. e., a reducible fraction
T*

T'

Original from
UNIVERSITY OF MICHIGAN

PRODUCTION OF MUSIC

405

Example:

T = 6t; T" = 4t

* -#
N 4t =3

Figure 6. A third illustration of duration-group distribution.

Example:

a a = 5a T = r s+2 = lOt ax = 6a

*

Figure 7. a a = 5a a T = 6a

(D* SiSl

(2) 6 attacks are equivalent to lOt; lOt X 5 = 50t

(3) When T" = f , = = 2 5T

p r 'I 8 i i r

ism

■ —

m

■ — ^ —

F — * 1 1 1 m^'F ]

:

a — *

Digiti

Figure S. T* = 25T (continued)

(^r\r\ri\{> Original from

VjUUJJU UNIVERSITY OF MICHIGAN

406

TECHNOLOGY OF ART PRODUCTION

gup*

r i J ' r i

i lip . | 1 - ,1f.

1 i , dg , m

-ffi — := EF z jf' a r === ~

1

u i »J i l r rj > J J i i

Figure 8. T" = 25T (concluded)

E. Synchronization of an Instrumental Group (pli) with an Attack-
Group (pla)

Example:

pli = 4; pla = 3; a a = 3 + 2 + 3 = 8; T = r s ^. 2 = lOt; 6a

(1) h Mil O) ¥ = ¥

(2) 8 X 4 = 32 (4) ^4 = -MP
(5) T" = 8t; W = ^; ^ = 20T*

■ V*

■ ■ ~i

^ —

liirrp

==F=

Figure 9. Synchronization of an instrumental group with an attack-group.

Example:

pli = 3; pla = 3; a a = 3 + 2 + 2 + 3 = 10; T = = 16t; 10a

(1) 1 = 1 (3) H = l

(2) 10-1 = 10 (4) 16-1 = 16

(5) T* = 8t; = 2T"

Original from

Digitized byGoOglG

UNIVERSITY OF MICHIGAN

PRODUCTION OF MUSIC

407

§ c _ r I r p r

li

Figure 10. pli = 3; pla = 3; a a = 3 + 2 + 2 + 3.

Example:

pli = 6; pla = 8; a a = r 5+4 = 20; T = = 16t; 10a T" = 8t

(1) PL =f=|; ||g| (3) A'=H = 6

(2) A = 20-3 = 60 (4) T = 16t-6 = 96t

(5) ^ = 12T"

(See Fig. 16, p. 411 for an example based on this formula)

Example of composition of the resultant of instrumental interference.

pli = pla = 2
Form of distribution: 5+3

Figure 11. pli = pla = 2

Digitized byGoOgle

5+3

Original from
UNIVERSITY OF MICHIGAN

408

TECHNOLOGY OF ART PRODUCTION

(1) f = 1

(2) 2 is an equivalent of 5 -f- 3 =8

(3) Duration -group: T = r 5+2 = 10t

a T = 6

8 _ 4 3 (8)
F - f 4 <6)

(4) lOt X 4 = 40t

(5) When T" = |, *f = ST*

Preliminary Scoring

Final Scoring

Figure 12. Preliminary and final score.

Example of composition of the resultant of instrumental interference.

pli = 3 pla = 3
Form of distribution: 8 4-3+5+2

Figure 13. pli = 3 ; pla = 3. Distribution: 8+3+5+2.

r\r\Ci\{> Original from

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PRODUCTION OF MUSIC 409

(1) 1 = 1

(2) 4 is an equivalent of 8+3+5+2=18

18 X 1 =18

(3) Duration group: r s+2 = lOt ^ = 3 3(6)

ax — 6

(4) lOt X 3 = 30t

(5) When T" = f , ^ = ^; L ^ A = 15T*

Preliminary Scoring

; rr
1 r

1 1 1 U E

r

err

r r

=F=-;

p p p p p

^ —

r

r

=F=-

Digits

Figure 14. Preliminary Score

Google

Original from
UNIVERSITY OF MICHIGAN

410

TECHNOLOGY OF ART PRODUCTION

Final Scoring

* r r r

\."

• a \ 1 LJ I

f=*. p

j r r

rtrrr

^ .-=

1 'Jl

m m m >j

* =*

.. ' ' '1

£= :

~ ^ _ - — • — )

: ~*V.

u.

r r

IV ■ ■ ■

Figure 15. Final score

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

PRODUCTION OF MUSIC

411

Example of composition of the resultant of instrumental interference.

pli = 6; pla'= 8;

Form of distribution: r s+ 4

Figure 16. pli ■= 6; pla = 8; r s+4

a) 1 = 1

(2) 8 is equivalent to 20 in r 5+4 20 X 3 = 60

(3) Duration-group = r^ a T = 10; f# - 6 6(10)

= 16t

(4) 16t X 6 = 96t; a given T* = f

(5) *g = 12T"

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

412

TECHNOLOGY OF ART PRODUCTION

Preliminary Scoring

w

P

P

p^p

r pr p

p

P

P

P«£

r p r p

p

Digili

Figure 17. Preliminary score.

(~~ , x , 1 1 , > Original from

lOOglC

UNIVERSITY OF MICHIGAN

PRODUCTION OF MUSIC
Final Scoring

413

CHAPTER 4
PRODUCTION OF KINETIC DESIGN

A. Proportionate Distribution within Rectangular Areas

1. Distributive squaring of the sum of two sides of a rectangle.

{ a b V _ a 2 + ab + ab + b*

\a+b a+b/~ (a + b) 2

ab

Areas:

I

II

IV

III

ab
b»

Figure 1. Distributive squaring of sum of two sides.

a 2 -ab ab-ab b 2 -ab a 2 -b 2
ab = : — — tz + ~, — — + : — r-rrr +

(a+b) 2 T (a+b) 2 T (a+b) 2 ^ (a+b) 2

_ a'b + a 2 b 2 + ab' + a 2 b 2
(a + b) 2

a 3 b a 2 b l ab» a 2 b 2

I = - — — ; II = - — — ; III = — ; IV =

(a+b) 2 ' (a+b) 2 ' (a+b) 2 ' (a+b) 2

Areas identical in square units: II and IV.

Areas non-identical in square units: I and III.

When a = b all areas are identical in square units and in form.

byt^OOgl 414 UNIVERSITY OF MICHIGAN

G

PRODUCTION OF KINETIC DESIGN

2. Distributive cubing of the sum of two sides of a rectangle.

a b V _ a' + a*b + a*b + ab* + a*b + ab* -f ab* + b 1

+ b a + b/ (a + b)»

a*b

a*b

ab 1

Areas:

I

II

V

VI

: IV

III

VIII

VII

XIII

XIV

IX

X

XVI

XV

XII

XI

a*b

ab 1

ab 1
b»

Figure 2. Distributive cubing of sum of two sides.

a»-a*b +a*b-a*b +a*b-ab* +a»-ab*

ab " £Tbv +

a*b • a*b + ab* • a*b + ab* • ab* + a 2 b • ab*
(a+b)»

a*b • ab 1 + ab 1 • ab* + ab* • b' + a*b • b»

i : . , . . "T

+

+

(a+b)»

a'-ab* + a*b-ab* +a*b-b» +a'-b' _
(a + b)»

(a'b + a'b* + a'b' + a 4 b*) + (a*b* + a»b» + a»b« + a'b')

(a+b)'

(a'b» + a*b 4 + ab 6 + a*b<) + (a 4 b* + a'b' + a*b« + a'b')

(a+b)'

+

Areas identical in square units:

II, IV, V, XIII. a*«b»

III, VI, VIII, IX, XIV, XVI a'b 1

VII, X, XII, XV a*b«

Areas non-identical in square units: I =a*b and XI = ab*.

r\r\Ci\f> Original from

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416

TECHNOLOGY OF ART PRODUCTION

3. Distributive involution of the fourth power of the sum of two sides of a rectangle.

+ a'b + a»b + a 2 b 2 + a 3 b + a 2 b 2 + a 2 b 2 + ab»

(

a + b a + b

Areas:

+

(a+b)«

a»b + a 2 b 2 + a 2 b 2 + ab» + a 2 b 2 + ab 3 + ab 3 + b*

(a+b)«

+

a'b

a'b

a'b*

a'b

i s b ! a ! b* ab*

I

II

V

VI

XVII

XVIII

XXI

XXII

IV

III

VIII

VII

XX

XIX

XXIV

XXIII

XIII

XIV

IX

X

XXIX

XXX

XXV

XXVI

XVI

XV

XII

XI

XXXII

XXXI

XXVIII

XXVII

XLIX

L

LI 1 1

LIV

XXXIII

XXXIV

XXXVII

XXXVIII

LII

LI

LVI

LV

XXXVI

XXXV

XL

XXXIX

LXI

LXI I

LVII

LVI 1 1

XLV

XLVI

XLI

XLII

LXIV

LXI 1 1

LX

LIX

XLVIII

XLVII

XLIV

XI. Ill

a'b

a'b*

a'b*

ab»

a'b*

ab»
ab»

b«

Figure 3. Distributive involution of fourth power of sum of two sides.

•a 2 b 2 + a<

ab = ir ^-a'b + a'b-a'b + a*b-a 2 b 2 + a«-a 2 b 2 ~|

IL (a -|- b)< J +

[" a'b-a'b + a 2 b-a»b + a 2 b 2 -a 2 b 2 + a*b-a 2 b 2 "|
+ L (a + b)« J

fa'b-a 2 b 2 + a 2 b 2 -a 2 b 2 + a 2 b 2 -ab» + a»b-ab»~|

*L &Tby J +

Digiti

(a + b)«

Google

Original from
UNIVERSITY OF MICHIGAN

PRODUCTION OF KINETIC DESIGN
+ a 3 b-a 2 b 2 + a'b-ab' + a 4 -ab» "j\

j~ a 4 -a 2 b 2

/[" a'b-a'b + a 2 b 2 -a 3 b + a^bj-a^ + a»b-a 2 b 2 ~|
U (a + b)« J +

+ f Vb^a'b + ab'-a'b + ab»-a 2 b 2 + a 2 b 2 -a 2 b 2 *j +

(a + b) 4

a 2 b 2 *a 2 b 2 + ab 3 -a 2 b 2 + ab'-ab 1

+ a 2 b 2 -ab»

(a+b) 4

a»b-a 2 b 2 + a 2 b 2 -a 2 b 2 + a 2 b 2 -ab

• + a 3 b-ab 3

+ L - r—rr. : ^ i +

L (a + b) 4 A)

jf" a 3 b-a 2 b* + a 2 b 2 -a 2 b 2 + a 2 b 2 -ab 3 + a 3 b-ab 3 "|
+ \l (a + by J

|~ a 2 b 2 -a 2 b 2 + ab 3 -a 2 b 2 + ab 3 -ab 3 + a 2 b 2 -ab* ~|
L (a + by J

f a 2 b 2 -ab 3 + ab 3 -ab 3 + ab 3 -b 4 + a 2 b 2 -b 4 ]
+ L ^+by J +

f a'b-ab 1 + a 2 b 2 -ab» + a 2 b 2 -b 4 + a 3 b-b 4 ~|\
L (a + b)* ](

j|~ a 4 -a 2 b 2 + a 3 b-a 2 b 2 + a 3 b-ab 3 + a 4 -ab 3 *]
U (a + b) 4 J

[~ a 3 b-a 2 b 2 + a 2 b 2 -a 2 b 2 + a 2 b 2 -ab 3 + a'b-ab' l
L (a+b) 4 J +

ra*b-ab* + a 2 b 2 -ab 3 + a 2 b 2 -b 4 + a 3 b-b 4 1
+ L (a~T^ J +

l~ a 4 -ab 3 + a 3 b-ab 3 + a 3 b-b 4 + a 4 -b 4 "|\
+ L (a+b) 4 Jf ~

_ j|~ (a 7 b + a'b 2 + a 5 b 3 + a'b 2 ) + (a 8 b 2 + a'b 3 + a 4 b 4 + a'b 3 )
" IL (a + b) 4

(a'b 3 + a 4 b 4 + a 3 b* + a 4 b 4 ) + (a'b 2 + a'b 3 + a 4 b 4 + a'b 3 ) ~|

(a+b) 4 J

|" (a«b 2 + a'b 3 + a 4 b 4 + a'b 3 ) + (a'b 3 + a 4 b 4 + a 3 b' + a 4 b 4 )

+

(a + b) 4

(a 4 b 4 + a'b 8 + a 2 b« + a 3 b') + (a'b 3 + a 4 b 4 + a'b 8 ,

(a + b) 4

> 8 + a 4 b 4 ) j

Digitized byGoOgle

Original from
UNIVERSITY OF MICHIGAN

418 TECHNOLOGY OF ART PRODUCTION

(a«b*

+ a*b*

+ a'b s

+ a 4 b 4 ) + (a 4 b 4

+ a»b s

+ a ! b 8

+ a'b 5 )

(a + b) 4

(a»b 6

+ ai i b t

+ ab 7

+ a«b«) + (a 4 b 4

+ a s b s

+ a s b«

+ a»b 6 )"

(a + b) 4

(a«b l

+ a»b»

+ a 4 b 4

+ a 5 b») + (a 6 b»

+ a 4 b 4

+ a»b 5

+ a 4 b 4 )

(a + b) 4

(a 4 b 4

+ a»b*

+ a'b 4

+ a»b*) + (a 8 b»

+ a 4 b 4

+ a»b 8

+ a 4 b 4 )

+

+ |i (TTbr +

*b>) + (a

(a + b) 4 J)

Areas identical in square units:

II, IV, V, XIII, XVII, XLIX a»b !

III, VI, VIII, IX, XIV, XVI, XVIII, XX, XXI, XXIX a»b l

XXXIII, L, LII.LIII.LXI

VII, X, XII, XV, XIX, XXII, XXIV, XXV, XXX, XXXII, a 4 b 4

XXXIV, XXXVI, XXXVII, XLV, LI, LIV, LVI, LVII, LXII, LXIV

XI, XXIII, XXVI, XXVIII, XXXI, XXXV, XXXVIII, XL, XLI, a»b 5

XLVI, XLVIII, LV, LVIII, LX, LXII I

XXVII, XXXIX, XLII, XLIV, XLVII, XLIX a ! b«

Areas non-identical in square units:
I = a*b and XLI 1 1 = ab 7

B. Distributive Involution in Linear Design

4:3

4 10° • ■ • O • • • 1 inch.

(3 + 1 +2-1-1 + 1+1 + 1+2 + 1+3) 1 = (9+3+6+3+3+3+3+6+3+9) +
(3+1+2 + 1+1+1+1+2 + 1+3) +(6+2+4+2+2+2+2+4+2+6) +
(3 + 1+2 + 1 + 1+1+1+2 + 1+3) +(3 + 1+2+1+1+1 + 1+2 + 1+3) +
(3+1+2+1+1+1+1+2 + 1+3) + (3 + 1+2+1+1+1+1+2+1+3) +
(6+2+4+2+2+2+2+4+2+6) + (3+1+2 + 1+1+1+1+2+1+3) +
(9+3+6+3+3+3+3+6+3+9)

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PRODUCTION OF KINETIC DESIGN

419

420

TECHNOLOGY OF ART PRODUCTION

PRODUCTION OF KINETIC DESIGN

421

Figure 6. Distributive involution in linear design {continued).

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TECHNOLOGY OF ART PRODUCTION

Figure 6. Lines of preceding configuration used as diameters of semi-circles (concluded.)

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PRODUCTION OF KINETIC DESIGN

423

C. Application to Dimensions and Angles

r (3-r2)

(2 + 1 + 1 + 2) 3
Z st = 5°. Direction : O

rv in *a k r^rinnlr* Original from

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TECHNOLOGY OF ART PRODUCTION

PRODUCTION OF KINETIC DESIGN

425

D. Positional Rotation Applied to Kinetic Design Within Rectangular
Area

<I> is equivalent to the proportionate rectangles of the third power.

Figure S. Positional rotation applied to kinetic design.

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426

TECHNOLOGY OF ART PRODUCTION

Figure 9. Positional rotation applied to kinetic design.

Digiti

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UNIVERSITY OF MICHIGAN

PRODUCTION OF KINETIC DESIGN

427

Figure 10. Positional rotation applied to kinetic design.

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Original from
UNIVERSITY OF MICHIGAN

428

TECHNOLOGY OF ART PRODUCTION

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CHAPTER 5

PRODUCTION OF COMBINED ARTS

A. The Time-Space Unit in Cinematic Design

IjClNETIC design for cinematic production deals with two elements:

1. the space of the screen

2. the time of the projection

The space of the screen may be used either as a conditionally limitless space,
or a definitely proportionate area with boundaries.

In treating the screen spate as a conditionally limitless area, it is necessary
to assign a certain linear unit which might be a simple partial area of the entire
screen. For example, it is possible to divide the length and the width of the screen
by 24, thus obtaining the units of horizontal and vertical directions, which through
their intersection, form boundary units of the same proportions as the entire
screen. The total number of such partial areas is 24 2 = 576. Such subdivision
permits a practically limitless number of designs, possible through various com-
binations of the partial area-units. If the number of designs to be executed is
more limited, one can limit the subdivision of the area of the entire screen into a
respectively smaller quantity of area-units.

Figure 1. The screen.

r^r^rslry Original from

lOU 8 K429 UNIVERSITY OF MICHIGAN

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TECHNOLOGY OF ART PRODUCTION

Treating the screen space as a definitely proportioned area with boundaries,
one has to adhere to its original harmony. The screen space of the silent film is a
4X3 rectangle, while the sound-track cuts off one side of it in such a way that
the remaining area becomes approximately 8X7. The problem of defining
partial area-units, harmonically related to the entire area of the screen, is as
fundamental as the problem of tuning in music. The solution of this problem
can be satisfied by the following formula:

+

a -I- b

a+b a+b a+b

= 1,

where a and b are the two sides of the screen area, or any rectangular area in
general.

Applying this formula to the silent screen, we obtain:

7 7 7

and for the sound screen:

+ ! =

15 15 .15

In order to develop the partial proportionate areas of the entire screen area,
it is necessary to subject the sum of two sides, as expressed above, to distributive
involution (squaring, cubing, etc.). The formula for distributive squaring,
through which we obtain 4 partial areas harmonically related to the original
area, is:

c-

+

;)'=

+

ab

+

ab

+ b ' a+b/ (a + b) 2 (a + b) 2 (a + b) 2
Applying this formula to the silent screen we obtain

(a+b) 2

\7 + 7/ 49

12 , 12 9

H

49 49 49

16

12

12

Figure 2. Distributive squaring of the screen.

Original from
UNIVERSITY OF MICHIGAN

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PRODUCTION OF COMBINED ARTS 431

Applying the same formula to the sound screen, we obtain:

\15 15/ ~ 225 225 + 225 ^ 225

64 56

56

49

Figure 3. Distributive squaring of sound screen.

Through application of the higher forms of involution through distributive
formulae similar to the one above, it is possible to split the entire area into any
desired number of partial areas. As squaring of the area produces 4 partial areas,
cubing produces 4 2 , i. e., 16 partial areas. Raising to the fourth power produces
4', t. e., 64 partial areas, and so on. Thus, all the partial areas are in harmony
with the entire area of the screen and serve as elements for the spatial cinematic
composition.

The time of projection must be approached on a similar basis. As the sound
film is projected at 24 frames per second, which at the same time is a desirable
speed for silent film, it is most practical to devise the individual time-units from
this number 24 as a group-unit of time measurement. Thus, the individual
units of time are projected single frames, which correspond to the individual
phases of the design made for animation: 24 drawings produce in projection one
second of fluent motion.

The distribution of the individual time-units (each equivalent to one phase
of drawing) in different quantities and groupings produces temporal rhythm.
Temporal rhythm must be coordinated with spatial rhythm in such a fashion that
one second of a screen projection, which is equivalent to 24 consecutive frames
of the film, must correspond to the formation of one linear or area-unit. For
example, in drawing a horizontal line through 24 consecutive phases, which
on the screen appears as a line growing to the length of one screen unit during
one second of projection, it is necessary for one phase to be gT* — tj-^TT °f tne
entire length of the screen. Thus, time and space can be coordinated on the basis
of one time-space unit, i.e., t = , second = 3^-5 of the screen length.

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TECHNOLOGY OF ART PRODUCTION

In making drawings for animation on a sheet 12" long, t = -fa". In projecting
a film made from such drawings on a screen 12' long, t becomes \ n in projection,
thus giving a 6" linear growth in the course of one second. This obviously guar-
antees a maximum graduality of motion.

B. Correlation of Visual and Auditory Forms

1. ELEMENTS OF VISUAL KINETIC COMPOSITION.

1. Linear, plane and solid trajectories

(distance, dimension, direction, form).

2. Illumination

(forms and intensity of light).

3. Texture

(density of matter, quality of surface).

4. General component: time.

2. ELEMENTS OF MUSIC

1. Frequency

(pitch).

2. Intensity

(relative dynamics).

3. Quality

(harmonic composition).

4. Density

(quantitative aggregation of sound).

5. General component: time.

The correlation of these two art forms may be performed through relative
coordination of the different components individually, and in groups, pertaining
to both art forms. All values must be determined from an initial axis-point or
line of the graph record of the composition.

distance
pitch + relative dynamics

distance

harmonic composition + quantitative
aggregation of sound

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distance

relative dynamics -+- harmonic com-
position

distance

quantitative aggregation of sound +
pitch

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PRODUCTION OF COMBINED ARTS

dimension dimension

pitch + relative dynamics

dimension

harmonic composition + quantitative
aggregation of sound

direction
pitch + relative dynamics

direction

harmonic composition + quantitative
aggregation of sound

form

pitch + relative dynamics
form

harmonic composition + quantitative
aggregation of sound

form of light
pitch -f- relative dynamics

form of light
harmonic composition + quantitative
aggregation of sound

intensity of light
pitch + relative dynamics

intensity of light
harmonic composition -f quantitative
aggregation of sound

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433

relative dynamics + harmonic com-
position

dimension

quantitative aggregation of sound -f
pitch

direction

relative dynamics + harmonic com-
position

direction

quantitative aggregation of sound +
pitch

form

relative dynamics + harmonic com-
position

form

quantitative aggregation of sound +
pitch

form of light

relative dynamics ■+- harmonic com-
position

form of light
quantitative aggregation of sound +
pitch

intensity of light
relative dynamics + harmonic com-
position

intensity of light
quantitative aggregation of sound +
pitch

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434 TECHNOLOGY OF ART PRODUCTION

density of matter density of matter

pitch + relative dynamics

density of matter

harmonic composition -f quantitative
aggregation of sound

quality of matter's surface
pitch -f- relative dynamics

quality of matter's surface
harmonic composition + quantitative
aggregation of sound

distance + dimension
pitch + relative dynamics

distance + dimension

harmonic composition + quantitative
aggregation of sound

dimension + direction
pitch + relative dynamics

dimension + direction
harmonic composition + quantitative
aggregation of sound

direction + form
pitch + relative dynamics

direction + form

harmonic composition + quantitative
aggregation of sound

Digitized byGoOgle

relative dynamics + harmonic com-
position

density of matter
quantitative aggregation of sound •+-
pitch

quality of matter's surface

relative dynamics + harmonic com-
position

quality of matter's surface

quantitative aggregation of sound +
pitch

distance + dimension
relative dynamics + harmonic com-
position

distance + dimension

quantitative aggregation of sound +
pitch

dimension + direction

relative dynamics + harmonic com-
position

dimension + direction
quantitative aggregation of sound +
pitch

direction + form

relative dynamics + harmonic com-
position

direction -f- form

quantitative aggregation of sound -f-
pitch

Original from
UNIVERSITY OF MICHIGAN

PRODUCTION OF

form + distance
pitch + relative dynamics

form + distance
harmonic composition + quantitative
aggregation of sound

forms + intensity of light
pitch + relative dynamics

forms + intensity of light
harmonic composition + quantitative
aggregation of sound

density of matter +
quality of matter's surface

pitch + relative dynamics

density of matter +
quality of matter's surface
harmonic composition + quantitative
aggregation of sound

distance + dimension
pitch

direction + form

pitch

distance + dimension
relative dynamics

direction + form
relative dynamics

distance -j- dimension

harmonic composition

direction + form

harmonic composition

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COMBINED ARTS 435

form + distance

relative dynamics + harmonic com-
position

form + distance
quantitative aggregation of sound -+■
pitch

forms + intensity of light

relative dynamics + harmonic com-
position

forms + intensity of light
quantitative aggregation of sound +
pitch

density of matter +
qualitv of matter's surface

relative dynamics + harmonic
composition

density of matter +
quality of matter's surface
quantitative aggregation of sound -f-
pitch

dimension + direction
pitch

form + distance
pitch

dimension + direction
relative dynamics

form + distance
relative dynamics

dimension + direction
harmonic composition

form + distance

harmonic composition

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UNIVERSITY OF MICHIGAN

436 TECHNOLOGY OF

distance ■+- dimension
quantitative aggregation of sound

direction + form
quantitative aggregation of sound

forms + intensity of light
pitch

forms 4- intensity of light
harmonic composition

density of matter +
quality of matter's surface
pitch

density of matter +
quality of matter's surface
harmonic composition

distance + dimension + direction
pitch

distance + dimension + direction
harmonic composition

dimension + direction + form
pitch

dimension + direction + form
harmonic composition

direction + form + distance
pitch

direction + form + distance
harmonic composition

form + distance + dimension
pitch

form + distance -f- dimension
harmonic composition

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ART PRODUCTION

dimension '+ direction
quantitative aggregation of sound

form + distance
quantitative aggregation of sound

forms + intensity of light
relative dynamics

forms + intensity of light
quantitative aggregation of- sound

density of matter +
quality of matter's surface
relative dynamics

density of matter +
quality of matter's surface
quantitative aggregation of sound

i

distance + dimension + direction
relative dynamics

distance + dimension + direction
quantitative aggregation of sound

dimension + direction + form
relative dynamics

dimension + direction + form
quantitative aggregation of sound

direction + form + distance
relative dynamics

direction + form + distance
quantitative aggregation of sound

form + distance -f dimension
relative dynamics

form + distance + dimension
quantitative aggregation of sound

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PRODUCTION OF COMBINED ARTS 437
distance + dimension + direction distance + dimension + direction

pitch + relative dynamics

distance + dimension -f direction
harmonic composition + quantita
tive aggregation of sound

dimension + direction + form
pitch + relative dynamics

dimension + direction + form

harmonic composition -f quantita-
tive aggregation of sound

direction -f form + distance
pitch + relative dynamics

direction + form + distance
harmonic composition -f quantita-
tive aggregation of sound

form + distance 4- dimension
pitch + relative dynamics

form -f distance + dimension
harmonic composition + quantita-
tive aggregation of sound

distance + dimension + direction
pitch + relative dynamics + har-
monic composition

distance + dimension + direction
harmonic composition + quantita-
tive aggregation of sound + pitch

Digitized byGoOgle

relative dynamics + harmonic com-
position

distance + dimension + direction

quantitative aggregation of sound
+ pitch

dimension + direction + form

relative dynamics + harmonic com-
position

dimension + direction + form

quantitative aggregation of sound
+ pitch

direction -f form + distance

relative dynamics harmonic com-
position

direction + form + distance
quantitative aggregation of sound
-f pitch

form + distance -+- dimension
relative dynamics + harmonic com-
position

form -f- distance ■+• dimension
quantitative aggregation of sound
+ pitch

distance + dimension + direction

relative dynamics + harmonic com-
position + quantitative aggregation
of sound

distance + dimension + direction
quantitative aggregation of sound
+ pitch + relative dynamics

UNIVERSITY OF MICHIGAN

438 TECHNOLOGY OF ART PRODUCTION

dimension + direction + form dimension + direction -+- form

pitch + relative dynamics + har-
monic composition

dimension -f direction + form

harmonic composition + quantita-
tive aggregation of sound + pitch

relative dynamics + harmonic com-
position + quantitative aggregation
of sound

dimension + direction + form
quantitative aggregation of sound
+ pitch + relative dynamics

direction + form + distance
pitch + relative dynamics + har-
monic composition

direction + form + distance

relative dynamics + harmonics com-
position + quantitative aggregation
of sound

distance + dimension + direction

pitch + relative dynamics + har-
monic composition

distance -f dimension -f direction
harmonic composition + quantita-
tive aggregation of sound + pitch

distance + dimension + direction

relative dynamics + harmonic com-
position ~f- quantitative aggregation
of sound

distance + dimension -f direction
quantitative aggregation of sound
+ pitch + relative dynamics

dimension + direction + form
pitch + relative dynamics + har-
monic composition

dimension + direction + form
harmonic composition + quantita-
tive aggregation of sound + pitch

dimension + direction -f form
relative dynamics + harmonic com-
position + quantitative aggregation
of sound

dimension + direction + form
quantitative aggregation of sound
+ pitch -f relative dynamics

direction + form -j- distance
pitch -f- relative dynamics + har-
monic composition

direction + form + distance

harmonic composition -f- quantita-
tive aggregation of sound + pitch

Digitized byGoOgle

direction + form 4- distance
relative dynamics + harmonic com-
position -f- quantitative aggregation
of sound

direction + form + distance
quantitative aggregation of sound
+ pitch -+- relative dynamics

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UNIVERSITY OF MICHIGAN

PRODUCTION OF COMBINED ARTS 439

form + distance + dimension form + distance + dimension

pitch + relative dynamics + har-
monic composition

form -f distance + dimension
harmonic composition + quantita-
tive aggregation of sound + pitch

relative dynamics + harmonic com-
position + quantitative aggregation
of sound

form -f distance + dimension

quantitative aggregation of sound
+ pitch + relative dynamics

distance

pitch + relative dynamics + har-
monic composition

distance

harmonic composition + quantita-
tive aggregation of sound + pitch

distance

relative dynamics + harmonic com-
position + quantitative aggregation
of sound

distance

quantitative aggregation of sound
+ pitch + relative dynamics

dimension

pitch relative dynamics + har-
monic composition

dimension

harmonic composition + quantita-
tive aggregation of sound + pitch

dimension

relative dynamics + harmonic com-
position + quantitative aggregation
of sound

dimension
quantitative aggregation of sound
+ pitch -f relative dynamics

direction

pitch + relative dynamics + har-
monic composition

direction

harmonic composition + quantita-
tive aggregation of sound + pitch

direction

relative dynamics + harmonic com-
position + quantitative aggregation
of sound

direction

quantitative aggregation of sound
+ pitch + relative dynamics

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TECHNOLOGY OF ART PRODUCTION
form form

440

pitch 4- relative dynamics + har-
monic composition

form

harmonic composition + quantita-
tive aggregation of sound + pitch

forms of light
pitch + relative dynamics + har-
monic composition

forms of light

harmonic composition -f quantita-
tive aggregation of sound + pitch

intensity of light
pitch 4- relative dynamics + har-
monic composition

intensity of light
harmonic composition + quantita-
tive aggregation of sound + pitch

density of matter
pitch + relative dynamics + har-
monic composition

density of matter
harmonic composition + quantita-
tive aggregation of sound + pitch

quality of matter's surface
pitch + relative dynamics + har-
monic composition

quality of matter's surface
harmonic composition + quantita-
tive aggregation of sound + pitch

Digitized byGoOgle

relative dynamics + harmonic com-
position + quantitative aggregation
of sound

form

quantitative aggregation of sound
+ pitch + relative dynamics

forms of light

relative dynamics + harmonic com-
position -f- quantitative aggregation
of sound

forms of light

quantitative aggregation of sound
+ pitch + relative dynamics

intensity of light

relative dynamics + harmonic com-
position + quantitative aggregation
of sound

intensity of light
quantitative aggregation of sound
4- pitch + relative dynamics

density of matter

relative dynamics + harmonic com-
position + quantitative aggregation
of sound

density of matter
quantitative aggregation of sound
+ pitch + relative dynamics

quality of matter's surface

relative dynamics 4- harmonic com-
position + quantitative aggregation
of sound

quality of matter's surface

quantitative aggregation of sound
4- pitch + relative dynamics

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UNIVERSITY OF MICHIGAN

PRODUCTION OF COMBINED ARTS 441

distance dimension

pitch + relative dynamics + har-
monic composition + quantitative
aggregation of sound

direction

pitch + relative dynamics + har-
monic composition + quantitative
aggregation of sound

forms of light
pitch + relative dynamics + har-
monic composition + quantitative
aggregation of sound

density of matter
pitch + relative dynamics + har-
monic composition -f- quantitative
aggregation of sound

pitch + relative dynamics + har-
monic composition + quantitative
aggregation of sound

form

pitch + relative dynamics + har-
monic composition -+- quantitative
aggregation of sound

intensitv of light
pitch + relative dynamics + har-
monic composition + quantitative
aggregation of sound

qualitv of matter's surface
pitch -1- relative dynamics + har-
monic composition + quantitative
aggregation of sound

distance + dimension

pitch + relative dynamics + har-
monic composition -+■ quantitative
aggregation of sound

direction + form

pitch + relative dynamics -f har-
monic composition -f quantitative
aggregation of sound

form + intensity of light
pitch + relative dynamics + har-
monic composition + quantitative
aggregation of sound

distance + dimension + direction
pitch -f relative dynamics + har-
monic composition + quantitative
aggregation of sound

direction + form -f distance

pitch + relative dynamics + har-
monic composition + quantitative
aggregation of sound

Digitized by GoOgle

dimension + direction
pitch -f- relative dynamics + har-
monic composition -f quantitative
aggregation of sound

form + distance
pitch + relative dynamics + har-
monic composition + quantitative
aggregation of sound

density of matter -f- quality of
matter's surface
pitch + relative dynamics + har-
monic composition + quantitative
aggregation of sound

dimension + direction + form
pitch + relative dynamics + har-
monic composition + quantitative
aggregation of sound

form + distance + dimension

pitch + relative dynamics + har-
monic composition -f- quantitative
aggregation of sound

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442 TECHNOLOGY OF ART PRODUCTION

The correlation of the general component in both art forms may be assigned
to different proportionate relations, such as harmonic ratios, distributive powers,
series of growth, etc. The entire manifold of synchronized components must be
based on a standard space-time unit expressed through a single motion picture
frame (^th of a second) and the common denominator of musical time.

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PRODUCTION OF COMBINED ARTS

443

Figure 5. One frame from a composition of kinetic design (cinema) with sound
(music), based on r a+7 in two reciprocally moving trajectories and on the diagonal
rotation of the background.

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TECHNOLOGY OF ART PRODUCTION

|i_L>MiNOSi-rr

| HUE

| SATURATION

LIGHT

DENSITY

WEIGHT

TEXTURE

| SIZE

| SHAPE

| POSITION

SffcTIAL

FORM

EXTENSION

D

U

R

A

T

1

N

SOUND

| TIM

BRE

PITCH

INTENSITY

Figure 6. Components of a combined kinetic art form.

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APPENDIX A.

BASIC FORMS OF REGULARITY AND COORDINATION 1

I. Binary and Ternary Synchronization
II. Distributive Involution — Groups
III. Groups of Variable Velocity

•These basic forms of regularity and coordina-
tion constitute a comprehensive elaboration of
formulae presented in Part Two, Theory of Regu-
larity and Coordination , Chapters 2-.S. (Ed.)

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oogle

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/. BIN A RY AND TERN A RY S YNCHRONIZA TION
OF GENETIC FACTORS {GENERATORS).

Nomenclature:

a — monomial periodic group (equivalent to sine-wave), representing major
genetic factor (major generator), whose phases have greater period than
that of—

Id — monomial periodic group (equivalent to sine-wave), representing minor
genetic factor (minor generator), whose phases have smaller period than
that of a.

a:b — a to b ratio

S — synchronization (coordination of durations), symmetrization (coordination
of extensions), the process in which a and b are coordinated by means
of complementary factors.

I — interference produced by the interaction of phases a and b.

r a+b — the resultant of interference of a to b, a harmonic-symmetric group,
possessing an axis of symmetry (therefore reversible), and belonging
to the basic forms of regularity.

The components of synchronization or symmetrization, including ab (the

product), -7- (the denominator), a, b and r, are applicable to:
ab

(1) identical components of one art-form;

(2) different components of one art-form;

(3) identical components of different art-forms;

(4) different components of different art-forms.

Formulae:

A. Binary Synchronization

a:b; S(a:b) = a-b = ab; S

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448

APPENDIX A

/. Rhythmic

Resultants

r 3+2 = 2 + 1+1+2

r« + , = 3 + 1+ 2+ 2 + 1+ 3

r i+2 = 2+ 2 + 1+1+2+2

r 6+3 =3+2 + 1+ 3 + 1+ 2+ 3

r i+ < = 4 + 1+ 3 + 2 + 2+ 3 + 1+ 4

r s+5 = 5 + 1 + 4 + 2+3+3 + 2+ 4 + 1 + 5

r 7+I = 2+ 2 + 2 + 1+1+2+2 + 2

r 7 , 3 = 3+ 3 + 1+ 2+3 + 2 + 1+ 3+ 3

r 7+ « = 4+ 3 + 1+ 4 + 2+ 2+ 4 + 1 +3+4

r 7+& = 5+ 2+ 3+ 4 + 1+ 5 + 1+ 4+3+2+5

r T+ i = 6 + 1+ 5+ 2+ 4+ 3+ 3+ 4 + 2+ 5 + 1+ 6

r^j = 3+ 3+ 2 + 1+3+3 + 1+ 2+3+3

r 8+& = 5+ 3 + 2+ 5 + 1 + 4+ 4 + 1+ 5+ 2+ 3+ 5

r 8 + 7 = 7 + 1+ 6 + 2+ 5+ 3+ 4 + 4+3+5+2+6 + 1 + 7

r 9+2 = 2+ 2+ 2 + 2 + 1+1+2+2+2+2

r 9+4 = 4+ 4 + 1+ 3+ 4 + 2 + 2+ 4+ 3 + 1+ 4+ 4

r»^5 = 5+ 4 + 1 + 5+ 3+ 2+ 5+ 2+ 3+ 5 + 1 + 4 + 5

r,+ 7 = 7+ 2+ 5+ 4+ 3+ 6 + 1+ 7 + 1 + 6 + 3+ 4 + 5+ 2 + 7

r 9+8 = 8 + 1+ 7+ 2+ 6+ 3+ 5+ 4+ 4 + 5+3+6 + 2+ 7 + 1+ 8

Digi

Google

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BINARY AND TERNARY SYNCHRONIZATION

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BINARY AND TERNARY SYNCHRONIZATION

451

452

APPENDIX A

Binary Synchronization

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BINARY AND TERNARY SYNCHRONIZATION

453

Binary Synchronization

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454

APPENDIX A

BINARY AND "TERNARY SYNCHRONIZATION 45 S

Binary Synchronization

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APPENDIX A

Biuarv Sy nchron izn lion

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APPENDIX A

B. Binary Synchronization with Fractioning
(Displacement of b through the phases of a).

a:b; S(a:b) = a-a = a 1 ; S ( — — - ) = — = — ;

\ a 3 b / a-a a*

- 1 {* it) +[*-•>+ »•» GO]} ■ where

NGb (the number of b groups) = a — b + 1;

hence: bit = at , b 2 t = ati, . . . b n to = at a _ b +i , where bi is the first Gb, bj is the
second Gb, etc., and where T , Ti, . . . represent the initial points of respective
phases (phasic origins or attacks).

1. Rhythmic Resultants with Fractioning
r Sj± = 2 + 1+1+1+1 + 1+ 2

r^ = 3 + 1+ 2 + 1+1+1+1+2 + 1+ 3

Tm = 2 + 2 + 8(1) + 1 + 8(1) + 2 + 2

= 3 + 2 + 1 + 2 + 4(1) + 1 + 4(1) +2 + 1+2 + 2

r^ = 4 + 1+ 3 + 1+1+2 + 1+ 2 + 1+1+3 + 1+ 4

r5± 5 = 5 + 1+ 4 + 1+1+3 + 1+ 2+ 2 + 1+ 3 + 1+1+4 + 1+ 5

r^ = 2 + 2 + 2 + 18(1) + 1 + 18(1) +2+2+2

r 7+ , = 3+ 3 + 1+ 2+1+2 + 12(1) + 1 + 12(1) + 2 + 1 + 2 + 1 +

+ 3+3

r2±i = 4 + 3 + 1+ 3 + 1+ 2 + 1+1+2 + 6(1) + 1 + 6(1) + 2 +

+1+1+2+1+3+1+3+4

r 1J± = 5+ 2+ 3+ 2+ 2 + 1+ 2+ 2 + 1 + 1+1+2 + 1+ 2 + 1 +

+1+1+2+2+1+2+2+3+2+5

r^ = 6 + 1+ 5 + 1+1+4 + 1

+ 1+5

Digitized byGoOgle

+2+3+1+3+2+1+4+1+
+ 1+6

Original from
UNIVERSITY OF MICHIGAN

BINARY AND TERNARY SYNCHRONIZATION 459

1-^ = 3+ 3 + 2 + 1+ 2 + 1+ 2 + 18(1) + 18(1) + 2 + 1 + 2 + 1 +

+2+3+3

rg^ = 5+ 3+ 2+ 3+ 2 + 1+ 2+ 2 + 1+ 2 + 1+1+1+1+2 +
+ 1+1+1+1+1+1+1+1+2 + 1+1+1+2 + 1 +
+ 2+ 2 + 1+ 2+ 3+ 2+ 3+ 5

rs^ = 7 + 1+ 6 + 1+1+5 + 1+ 2+ 4 + 1+ 3+ 3 + 1+ 4 + 2 +
+1+5+1+1+6+1+7

r 9±L = 2 + 2 + 2 + 2 + 32(1) + 1 + 32(1) + 2 + 2 + 2 + 2

r 9j± = 4+ 4 + 1+ 3 + 1+ 3 + 1+1+2 + 1+1+2 + 16(1) + 1 +
+ 16(1) + 2 + 1+1+2 + 1+1+3 + 1+ 3 + 1+ 4 + 4

1-^=5*+ 4 + 1+ 4 + 1+ 3 + 1+1+3 + 1+1+2+1+1+1 +
+ 2 + 8(1) + 1 + 8(1) + 2 + 1+1+1+2 + 1+1+3 + 1 +
+1+3+1+4+1+4+5

rj!±I = 7+ 2+ 5+ 2+ 2+ 3+ 2+ 2+ 2 + 1+ 2+ 2+ 3 + 1+1 +

+ 2+ 3 + 2 + 1+1+3+2+2 + 1+ 2 + 2+ 2+ 3+ 2+ 2 +
+5+2+7

r?± 8 = 8 + 1+ 7 + 1+1+6 + 1+ 2+ 5 + 1+ 3+ 4 + 1+ 4 + 3 +
+1+4+3+1+5+2+1+6+1+1+7+1+8

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

V

460

APPENDIX A

BINARY AND TERNARY SYNCHRONIZATION 461

Binary Synchronization with jractioning

r\r\ri\(> Original from

«J by UNIVERSITY OF MICHIGAN

462

APPENDIX A

BINARY AND TERNARY SYNCHRONIZATION

463

464

APPENDIX A

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

BINARY AND TERNARY SYNCHRONIZATION

46S

466

APPENDIX A

BINARY AND TERNARY SYNCHRONIZATION 467

468

APPENDIX A

470

APPENDIX A

APPENDIX A

BINARY AND TERNARY SYNCHRONIZATION 473

474 APPENDIX A

BINARY AND TERNARY SYNCHRONIZATION 47S

476

APPENDIX A

C. Ternary Synchronization

Ternary synchronization includes 3 generators: a, b and c, their product
(abc), their denominator ^^^> their complementary factors (be, ac, ab) and two
resultants of interference: r and r'.

r represents the resultant of interference of the generators; it has a number of
recurrence-groups ;

r' represent the resultant of interference of the complementary factors; it is of
the type r (a+b) .

r

— produces a perfect coordination of two contrasting forms of regularity, each
r

accompanied by its own generators.

The presence of two resultants is characteristic of all polynomials.

a:b:c ; S(a:b:c) = a-b'C = abc ; s( — - — ) = -7-;

\a:b:c/ abc

r,. +b «, - 1 b° (£) * ac Ob) * ab (£)]

+ "(=)*'(=)]•

D. Generalization: Synchronization of n Generators.

a:b:c: ... : m; S(a:b:c : . . . : m) = abc . . . m ;

s( ! ) !—

\a:b:c: . . . : m/ abc . . . m

) = I I bed . . . m ( — — ) -h acd . . . m ( — ) -5-

L Vabc . . . m/ \abc . . . m/

t- abd ...ml - — ) . . . -b abc ... 1 ( - — — ) ;

\abc . . . m/ Vabc . . . m/ J

_ r / bed . . . m\ / acd . . . m \
m) L Vabc . . . m/ Vabc . . . m/

/a^\ / abc... 1 \1

vabc . . . m/ \abc . . . m/ J

n' -■ _j l r\r\cs\f> Original from

Digilize<J by ^UU^IC UNIVERSITY OF MICHIGAN

r (a + b+c+ . . . +m

1 (a+b+c+.

BINARY AND TERNARY SYNCHRONIZATION

477

1. Rhythmic Resultants from Three Generators

r J+s+6 = 2 + 1 + 1+1+1+2 + 1+1+2+2+14-1+2 +
+2+1+1+2+1+1+1+1+2

6+4+2+3+3+2+4+6

r, +5+8 = 3 + 2 + 1+ 2 + 1+1+2+3 + 1+ 2+ 2 + 1+ 3 +
+ 1+ 2+ 3+ 2 + 1+ 2 + 1+ 3 + 1+ 2+ 3+ 3 +
+ 2 + 1+ 3 + 1+1+1+3+3 + 1+1+1+3 +
+ 1+ 2+ 3+ 3+ 2 + 1+ 3 + 1+ 2 + 1+ 2+ 3 +
+2+1+3+1+2+2+1+3+2+1+1+2
+1+2+3

r', +i+8 = 15+9 + 6 + 10 + 5+ 3 + 12 + 12+3 + 5 + 10 + 6 + 9+15

rn-n-ii = 3 + 2 + 1+ 3 + 1+ 2 + 1+ 2+ 3+ 2 + 1+ 3 + 1 +

+ 1 + 1+ 3+ 3+ 2 + 1+ 3 + 1+ 2+ 3+ 3+ 2 + 1 +
+ 1+ 2 + 1+ 2+ 3+ 3+ 2 + 1+ 3 + 1+ 2+ 3+ 3 +
+ 2 + 1+ 3 + 1+ 2+ 3 + 1+ 2 + 2 + 1+ 3 + 1+ 2 +
+ 2 + 1+ 3+ 2 + 1+ 3 + 1+ 2+ 3+ 3+ 2 + 1+ 3 +
+ 1+ 2+ 3+ 3+ 2 + 1+ 2 + 1 + 1+ 2+ 3+ 3+ 2 +
+ 1+ 3 + 1+ 2+ 3+ 3 + 1+1+1+3 + 1+ 2+ 3 +
+2+1+2+1+3+1+2+3

• r'^n = 15+ 15+9+6 + 15+5 + 10+3 + 12 + 15 + 12+3 +
+ 10 + 5 + 15+6+9 + 15 + 15

r 6+8+18 = 5+ 3 + 2+ 3 + 2 + 1+ 4+ 4 + 1+1+4 + 2+ 3 +
+ 4 + 1+ 5+ 3+ 2+ 2+ 3 + 1+ 4 + 4 + 1+ 5 +
+ 2+ 3+ 3+ 2+ 5+ 3+ 2 + 1+ 4 + 1+ 4+ 4 +
+ 1+ 5+ 2+ 3+ 2+ 3+ 5+ 3+ 2+ 5 + 1+ 4 +
+ 3 + 1 + 1+ 5+ 2+ 3 + 1+ 4 + 5+ 3 + 1 + 1 +
+ 5 + 1+ 4 + 2 + 2 + 1+ 5+ 2+ 3+ 5+ 5+ 3 +
+ 2+ 5+1+4 + 1+ 3 + 1+ 5+ 2+ 2 + 1+ 5 +
+ 5+ 2 + 1+ 2+ 5 + 1+ 4 + 4 + 1+ 5+ 2 + 1 +
+ 2+ 5+ 5 + 1+ 2+ 2+ 5 + 1+ 3 + 1+ 4 + 1 +
+ 5+ 2+ 3+ 5+ 5+ 3+ 2+ 5 + 1+ 2+ 2+ 4 +
+ 1+5 + 1+1+3+5+4 + 1+3+2+5 + 1 +
+ 1+ 3+ 4 + 1+ 5+ 2+ 3+ 5+ 3+ 2+ 3+ 2 +
+ 5 + 1+ 4+ 4 + 1+ 4 + 1+ 2+ 3+ 5+ 2+ 3 +
+3+2+5+1+4+4+1+3+2+2+3+5+
+1+4+3+2+4+1+1+4+4+1+2+3+
+2+3+5

Digil

oogle

Original from
UNIVERSITY OF MICHIGAN

APPENDIX A

40 + 25 + 15 + 24 + 16 + 10 + 30 + 35 + 5 + 8 +
+ 32 + 20 + 20 + 32 + 8 + 5 + 35 + 30 -f 10 + 16 +
4- 24 + 15 + 25 + 40

3+1+2+1+1+1+3+2+1+1+2+2+
+1+3+3+1+2+2+1+2+1+3+1+2+
+2+1+3+1+2+1+2+2+1+3+3+1+
+2+2+1+1+2+3+1+1+1+2+1+3

12+9+3+4 + 8+ 6+ 6+ 8+ 4+ 3+ 9 + 12

3+1+2+2+1+2+1+3+1+2+2+1+

+1+2+3+1+2+2+1+3+3+1+2+

+2+1+3+3+1+2+1+1+1+3+3+

+ 1+ 2+2 + 1+ 3+3 + 1+1+1+2 + 1 +

+3+3+1+2+2+1+3+3+1+2+2+

+1+3+2+1+1+2+2+1+3+1+2+

+1+2+2+1+3

12 + 12+9+3+8+4 + 12+6 + 6 + 12+4 +
+ 8+ 3+ 9 + 12 + 12

3+3+1+2+2+1+2+1+3+3+1+2+
+3+1+2+3+2+1+3+3+2+1+3+
+ 1+ 2+ 3 + 1+1+1+3+3+3+3 + 1 +
+ 2+ 3 + 2 + 1+ 3+ 3+ 3 + 1+ 2 + 1+ 2 +
+3+2+1+3+3+3+2+1+1+2+3+
+2+1+1+2+3+3+3+1+2+3+2+
+ 1+ 2 + 1+ 3+ 3+ 3 + 1+ 2+ 3+ 2 + 1 +
+ 3+ 3+ 3+ 3 + 1+1+1+3+2 + 1+ 3 +
+ 1+ 2+ 3+ 3 + 1+ 2+ 3+ 2 + 1+ 3+ 2 +
+1+3+3+1+2+1+2+2+1+3+3

21 + 12 + 9 + 21 + 3 + 11 + 7 + 15 + 6 + 21 +
+ 6 + 15+7 + 11+3+21+9 + 12+21

Google

Original from
UNIVERSITY OF MICHIGAN

BINARY AND TERNARY SYNCHRONIZATION

479

1 + 2 + 2 +

4 +

1 +

1 + 2 + 4 +

+ 4 + 2 +-2

i

~r

A

+ 1

+ 3 + 3 +

+2+2+2

i

-r

L

+ 4

+ 1 + 3 +

+4+2+1

i

-r

1

1

+ 4

+ 1 + 3 +

+1+1+3

i

+

L

+ 2

+ 4 + 1 +

+3+1 +4

+

2

+ 2

+ 4 + 1 +

+3+1+4

+

2

+ 2

+ 3 + 1 +

+2+3+1

+

4

+ 1

+ 1 +2 +

+4+3+1

+

4

+ 2

+ 2 + 2 +

+1+3+3

+

1

+ 4

+ 2 + 2 +

+4+4+2

+

1

+ 1

+ 4 + 2 +

+2 + 1+ 3 + 1+ 3+ 4

r 4+7+m

= 28 + 16 + 12 + 21 + 7 + 4 + 24 + 20 + 8 + 14 +
+ 14 + 8 + 20 + 24 + 4 + 7 + 21 + 12 + 16 + 28

r 4 +n., = 4 + 1+ 3 + 1+1+2+3 + 1+ 2+ 2+ 4 + 1 +
+ 2 + 1+ 2+ 2+ 3 + 1+ 4 + 4 + 1+ 3+ 2 +
+ 2 + 2 + 1+1+4 + 3 + 1+1+3+2 + 2 +
+ 3 + 1+ 4 + 1+ 3 + 1+ 3 + 2+ 2+ 3 + 1 +
+ 3 + 1+ 4 + 1+ 3+ 2 + 2+ 3 + 1+1+3 +
+ 4 + 1+1+2 + 2+ 2+ 3 + 1+ 4+ 4 + 1 +
+ 3 + 2+ 2 + 1+ 2 + 1+ 4 + 2+ 2 + 1+ 3 +
+2+1+1+3+1+4

r'4+6+» = 20 + 16+4 + 5 + 15 + 12+8 + 10 + 10 + 8 +
+ 12 + 15+5+4 + 16 + 20

4 +

1 +

3 +2 +2 +2 +

1 +

1 +

4 +4 + 1 +3 +

+ 2

+

2

+3+1+4+2

+ 2

+

1

+ 3 +2 +2 +

+ 3

+

1

+4+4+1+3

+ 2

+

2

+ 3 + 1 +4 +

+ 4

+

1

+3+2+2+3

+ 1

+

2

+ 2 +4 + 1 +

+ 3

+

2

+2+3+1+4

+ 4

+

1

+ 1 + 2 +2 +

+ 2

+

3

+1+4+4+1

+ 3

+

2

+ 2 +2 + 1 +

+ 1

+

4

+4+1+3+2

+ 2

+

3

+ 1 +4 + 2 +

+ 2

+

1

+ 3 + 2 + 2,+ 3

+ 1

+

4

+ 4 + 1 +3 +

+ 2

+

2

+3+1+4+4

+ 1

+

3

+ 2 +2 +3 +

+ 1

+

2

+2+4+1+3

+ 2

+

2

+ 3 + 1 +4 +

+ 4

+

1

+1+2+2+2

+ 3

+

1

+ 4

rv -■ _j l r\rscs\{> Original from

Digitized by VjUUJJU. UNIVERSITY OF MICHIGAN

480 APPENDIX A

r' iM < = 20 + 20 + 16 + 4 + 10 + 10 + 20 + 12 + 8 + 20 +
+ 20 + 8 + 12 + 20 + 10 + 10 + 4 + 16 + 20 + 20

r i+i+u = 4 + 4 + 1+ 3+ 2+ 2+ 2+ 2+ 4+ 3 + 1+ 4 +

+

4

+

4

+

2

+

2

+

1

+ 3

+

4

+

2

+

2

+ 4

+ 3

+

+

1

+

4

+

2

+

2

+

4

+ 4

+

1

+

3

+

4

+

I

+ 2

+

+

4

+

2

+

1

+

1

+

4

+ 4

+

4

+

4

+

1

+

-J

6

+ 4

+

+

2

+

2

+

4

+

3

+

1

+ 4

+

4

+

4

+

4

i

+

1

+ 1

+

+

2

+

4

+

2

+

2

+

4

+ 3

+

1

+

4

+

4

+

2

+ 2

+

+

4

+

1

+

3

+

4

+

2

+ 2

+

4

+

3

+

1

1

+

2

+ 2

+

+

4

+

4

+

4

+

1

+

3

+ 4

+

2

+

2

+

2

+

2

+ 3

+

+

1

+

4

+

4

+

4

+

4

+ 1

+

3

+

2

+

2

+

2

+ 2

+

+

4

+

3

+

1

+

4

+

4

+ 4

+

2

+

2

+

1

+

3

+ 4

+

+

2

+

2

+

4

+

3

+

1

+ 4

+

2

+

2

+

4

+

4

+ 1

+

+

3

+

4

+

2

+

2

+

4

+ 2

+

1

+

1

+

4

+

4

+ 4

+

+

4

+

1

+

3

+

4

+

2

+ 2

+

4

+

3

+

1

+

4

+ 4

+

+

4

+

4

+

1

+

1

+

2

+ 4

+

2

+

2

+

4

+

3

+ 1

+

+

4

+

4

+

2

+

2

+

4

+ 1

+

3

+

4

+

2

+

2

+ 4

+

+

3

+

1

+

2

+

2

+

4

+ 4

+

4

+

1

+

3

+

4

+ 2

+

+

2

+

2

+

2

+

3

+

1

+ 4

+

4

= 36 + 20 + 16 + 36 + 4 + 14 + 18 + 24 + 12 +
+ 36 + 8 + 28 + 28 + 8 + 36 + 12 + 24 + 18 +
+ 14 + 4 + 36 + 16 + 20 + 36

Digitized by GoOgle

BINARY AND TERNARY SYNCHRONIZATION

481

r s+9+I4 = 5+ 4 + 1+ 4 + 1+ 3 + 2+ 5+ 2 + 1+ 2+ 5 +
+ 1+ 4 + 2+ 3+ 5+ 4 + 1+1+4 + 3+ 2 +
+5+2+3+5+1+3+1+5+5+3+1+
+ 1+ 5+ 3+ 2+ 2+ 3 + 2+ 3+ 5 + 1+ 4 +
+ 5+ 5+ 4 + 1+ 5+ 3 + 1 + 1+ 5+ 2+ 3 +
+3+2+1+4+5+2+3+4+1+5+1+
+2+2+5+2+3+5+1+4+4+1+5+
+4+1+3+2+3+2+5+2+3+5+1+
+4+1+4+5+4+1+5+3+2+4+1+
+ 2+ 3+ 5 + 1+ 2+ 2+ 5+ 5+ 2+ 2 + 1 +
+ 5+ 3+ 2 + 1+ 4 + 2+ 3+ 5 + 1+ 4 + 5 +
+ 4 + 1+ 4 + 1+ 5+ 3+ 2+ 5+ 2+ 3+ 2 +
+ 3 + 1+ 4 + 5 + 1+ 4+ 4 + 1+ 5+ 3+ 2 +
+5+2+2+1+5+1+4+3+2+5+4+
+1+2+3+3+2+5+1+1+3+5+1+
+4+5+5+4+1+5+3+2+3+2+2+
+ 3+ 5 + 1+1+3+5+5 + 1+ 3 + 1+ 5 +
+3+2+5+2+3+4+1+1+4+5+3+
+2+4+1+5+2+1+2+5+2+3+1+
+4+1+4+5

r' M+u = 45 + 25 + 20 + 36 + 9 + 5 + 40 + 30 + 15 + 27 +
+ 18 + 10 + 35 + 35 + 10 + 18 + 27 + 15 + 30 +
40 + 5 + 9 + 36 + 20 + 25 + 45

Digitized by GoOgle

482

APPENDIX A

Trinomial Synchronization

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

BINARY AND TERNARY

SYNCHRONIZATION

483

BINARY AND TERNARY SYNCHRONIZATION

485

484

APPENDIX A

486

APPENDIX A

4S8

APPENDIX A

BINARY AND TERNARY SYNCHRONIZATION

489

490

APPENDIX A

BINARY AND TERNARY SYNCHRONIZATION

491

492

APPENDIX A

BINARY AND TERNARY SYNCHRONIZATION 493

494

APPENDIX A

Figure 56, Ternary Synchronization (continued)

Digitized by ^.OOglC UNIVERSITY OF MICHIGAN

BINARY AND TERNARY SYNCHRONIZATION

495

Figure 56. Ternary Synchronization ( concluded j

Digitized by Google UNIVERSITY

496

APPENDIX A

BINARY AND TERNARY SYNCHRONIZATION

497

498

APPENDIX A

BINARY AND TERNARY SYNCHRONIZATION 499

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4
4

/l 2 lV

DISTRIBUTIVE INVOLUTION GROUPS 569
1. Cube of Trinomials

(l+2+l)+(2+4+2)+(l+2+l)+(2+4+2)+(4+8+4) +

64

+(2+4+2) + (l+2 + l) + (2+4+2)+(l+2 + l)

\4 4 4/

64

(8+4+4)+(4+2+2)+(4+2+2) + (4+2+2)+(2 + l+l) +

64

+<2+l+l) + (4+2+2)+(2+l+l)+(2 + l+l)

(I+1+2Y = <I±i

\4 4 4/

64

+2) + (1 +1 +2) + (2+2+4) +(1 +1 +2) +(1 + 1 +2) +

r» =

64

-f (2 +2 +4) +(2+2 +4) +(2 +2 +4) +(4+4+8)

64

6(l)+2+2+2+l+2+3(l)+2+l+l+10(2)+l+l+2+3(l)+2 + l +

64

+2+2+2+6(1)
64

(4+8+4)+(8 + 16+8)+(4+8+4)
64

(16+8+8)+(8+4+4) + (8+4+4)
64

(4+4+8)+(4+4+8)+(8+8 + 16)

4\4 4 4/

*(?+MY -

4\4 4 4/
4\4 4 4/

r" =

64

6(4) +8 +8 +6(4)

64

16/1 2 l\ _ 16+32 + 16

16\4 4 4/ ~ 64

16/2 1 l\ _ 32 + 16 + 16

16\4 4 4/ ~ 64

16/1 1 2\ _ 16 + 16+32

16\4 4 4/

r =

64

16 + 1 6 + 16+16
64

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

570 APPENDIX A

5
5

(l+3 + l)+(3+9+3) + (l+3+l)+(3+9+3)+(9+27+9) +

125

+(3+9+3)+(l+3+l)+(3+9+3)+(l+3 + l)

125

(27+9+9)+(9+3+3)+(9+3+3)+(9+3+3)+(3+l+l) +

125

+(3+l+l)+(9+3+3)+(3+l+l) + (3 + l+l)

125

+3) + (l+l+3)+(3+3+9)+(l+l+3) + (l+l+3) +

125

+ (3 +3 +9) + (3 +3 +9) + (3 +3 +9) + (9 +9 +27)

125

, = l+l+2+4(l)+2+3+3 + l+3 + l+3 + 4(l)+2 + l+l+3 + l+l+l +
f 125

+2 + l+4+4+l+3 + l+2 + l+3+5+3 + l-|-2 + l+3+l+4+4-fl-|-2-|-

125

+ 1+1+1 +3 + 1 +1+2+4(0+3+1 +3 + 1 +3+3+2+4(l)+3+l+l

125

(8+4+8) + (4+2+4)+(8+4+8) + (4+2+4)+(2 + l+2) +

(2 1 2\»

(5+5+5) '

(5+i+i) =

(H+D"

125

+(4+2+4)+(8+4+8)+(4+2+4) + (8+4+8)

125

(l+2+2)+(2+4+4)+(2+4+4)+(2+4+4)+(4+8+8) +

125

+(4+8+8)+(2+4+4)+(4+8+8)+(4+8+8)

125

_ (8+8+4)+(8+8+4)+(4+4+2)+(8+8+4)+(8+8+4) +

125

+(4+4+2)+(4+4+2)+(4+4+2)+(2+2 + l)

125

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS 571

^ (continued)

, = 1+2+2+2 + 1 +3 + 1 +3 + l+l+3 + l+3+4(l)+2+l+4+l +2 + 1 +
r 125

+ 1+2+2+3 + 1+2+4+1+1+2 + 1+1+2 + 1+2+1+1+2+1+1+4+

125

+2 + 1 +3+2+2 + 1+1 +2 + 1+4+1 +2 + 4(l)+3 + l+3+l+l +3 + 1 +

125

+3 + 1+2+2+2 + 1
125

l+45 + 15)+(5+15+5)

5/1 3 lV = (5 + 15+5)+(15-
5\5 5 5/

5/3 1 lV = (45 + 15 + 15)+(15
5\5 5 5/

HW-4)' =

5\5 5 5/

125

+5+5)+(15+5+5)

125

(5+5+15)+(5+5 + 15)+(15 + 15+45)

r l =

125

5+5 + 10+6(5) +10+5 + 10+6(5) +10+5+5

125

5/2 1 2V _ (20 + 10+20) +(10+5 + 10) +(20+10+20)
5\5 + 5 + 5/ " 125

(5 + 10 + 10) + (10+20+20) +(10+20+20)
125

(20+20 + 10) +(20+20 + 10) +(10 + 10+5)
125

1(1-1-!)' -

5\5 5 5/

5 + 10+5(5) + 10+5(5) + 10+5(5) + 10+5

r» =

25/1 3 l\ _ 25+7
25\5 + 5 5/ ~ 1

125
+75+25

25/3 1 l\ _ 75+2
25\5 + 5 5/ " 1

25

+25+25

25/1 1 2\ _ 25+2
25\5 + 5 + 5/ ~ 1

25 '
+25+75

r =

125

25+25 +25+25+25
125

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

572

APPENDIX A

5

- (continued)

25/2 1 2\ _ 50+25+ 50
25\5 5 5/ ~ 125

'l 2 2\ _ 25+50+50
^ + 5 + 5/ ~

125

25/2 2 A _ 50+50+25
25\5 5 5/ ~ 125

25+25+25+25+25

r —

125

(27+9+27)+(9+3+9) + (27+9+27) + (9+3+9) +

343

+ (3 + 1 +3) + (9+3+9)+(27 +9+27) +(9+3+9)+(27+9+27)

343

(l+3+3)+(3+9+9)+(3+9+9) + (3+9+9) + (9+27+27) +

343

+ (9+27 +27) + (3+9+9) +(9+27+27) +(9+27+27)

343

(27+27+9) + (27+27 +9) +(9+9+3) +(27+27+9) +

r« =

343

+(27+27+9)+(9+9+3)+(9+9+3)+(9+9+3)+(3+3+l)

343

1+3+3+3+9+8+1+3+5+4+9+3+2+7+2+7+2+3+4+5 +

343

+6 + 16+5+6+3+6 + 7+2+7+2+3+9+3+9 + 1+2 + 1+2 + 1+9 +

343

+3+9+3+2+7+2+7+6+3+6+5 + 16+6+5+4+3+2+7+2 +

343

+7+2+3+9+4+5+3 + 1+8+9+3+3+3 + 1

343

Digitized byGoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS

573

7

- {continued)

(2 3 2V

(8 + 12+8) +(12 + 18 + 12) + (8 + 12+8)+(12 + 18 + 12) +

343

+ (18+27 + 18) +(12 + 18 + 12) +(8 + 12 +8) +(12 + 18 + 12) +

343

+ (8 + 12+8)

/3 2 2V

(7+7+7) "

343

(27 + 18 + 18) +(18 + 12 + 12) +(18 + 12 + 12) +(18 + 12 + 12) +

(2 2 l\

(7+7+7) "

343

+ (12+8+8)+(12+8+8) +(18 + 12 + 12) +(12+8+8) +

343

+ (12+8+8)
343

(8+8 + 12) + (8+8 + 12) + (12 + 12 + 18) +(8+8 + 12) +

343

+ (8+8 + 12) + (12 + 12 + 18) + (12 + 12 + 18) +(12 + 12 + 18) +

__

+ (18 + 18 + 27)
343

= 8+8+4 + 7 + 1+8+4+4 + 1+11+2+5+5+2+8+2 + 1+9+3+5 + 7 +

343

+ 1+4+4+9+3 + 2+6 + 1+5+2+5 + 7+4 + 7 + 1+11+1+7+4 + 7+5 +

343

+ 2+5 + 1+6+2+3+9+4+4 + 1+7+5+3+9 + 1+2+8+2+5+5+2 +

343

+ 11 + 1+4+4+8 + 1+7+4+8+8

7/3 1 3V

7(7+7+7) =

7/1 3 3V

h +7 +7) -

7/3 3 l\ 2

7(7+7+7) -

343

63 + 21+63+21+7+21+63+21+63
343

7+21 +21 +21 +63+63 + 21 +63+63
343

63+63+21+63+63+21+21+21+7

343

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

574 APPENDIX A

7

- (continued)

r s =

7+21+21+14+7 + 14+42+7 + 14+21+7+21+14 + 7+42 + 14 + 7 +

343

+ 14+21+21+7
343

28 +42 +28 +42 +63 +42 +28 +42 +28

7 -(-+-+-)' -

7\7 7 7/
7\7 7 T 7/
7\7 7 7/

343

63 +42 +42 +42+28+28 +42+28+28
343

28+28 +42 + 28+28 +42 +42 +42 +63

r 2 =

343

28+28+7+7+28+7+21+14+7+7+35+7+7 + 14+21+7+28+7 +

343

+7+28+28

343

±?(V+ 3 ~)

147+49 + 147

343

49/1 3 3\ 49 + 147 + 147

)\7 7^7/

49V7 7 7/ 343

49/3 3 A _ 147 + 147+49
49\7 7 7/ ~ 343

_ 49+98+49+98+49
f ~ 343

49/2 3 2\ 98 + 147+98
49\7 7 + 7/ ~ 343

147+98+98
343

98+98 + 147

'2 3 2\

,7+7+7) "

49/3 2 2\

79(7+7+7) "

49\7 7 7/

r =

343

98+49+49+49+98
343

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS 575

8
8

( 3 -+ 2 -+ 3 )' =

\8 8 8/

(2 3 3V

\8^8 8/

(27 + 18+27) +(18 + 12 + 18) + (27+18+27) + (18 + 12+18) +

512

+ (12+8 + 12) +(18 + 12 + 18) +(27+18+27) +(18 + 12 + 18 ) +

512

+ (27 + 18+27)
512

(8 + 12 + 12) +(12 + 18 + 18) +(12 + 18-4-18) +(12 + 18 + 18) +

512

+(18+27 +27) + (18+27+27) +(12 + 18 + 18) +(18+27 +27) +

512

+ (18+27+27)
512

(27 +27 + 18) + (27 +27 +18) + (18 + 18 + 12) + (27 +27 + 18) +

512

+ (27 +27 + 18) +(18 + 18 + 12) +(18+18 + 12) +(18 + 18+12) +

512

+ (12 + 12+8)
512

. 8 + 12 + 7+5 + 12 + 1+9+8 + 10+8 + 10+2 + 7+3+8 + 10+6+2 + 12 +

r s =

512

4+3 + 11+4+3 + 11+4 + 12+2 + 16+9+2 + 1+18+6+2+4+8+4+2 +

512

+6 + 18 + 1+2+9 + 16+2 + 12+4 + 11+3+4 + 11+3+4 + 12+2+6 +

512

+10+8+3+7+2 + 10+8 + 10+8+9 + 1+12+5+7 + 12+8

512

+48+32+48+72+48+72

8/3 2 3\ 2 _ 72+48 + 72
8\8 + 8 _l "8/

?(WY =

8\8 8 8/

512

32+48+48+48+72+72+48+72+72
512

+ 3 2 V 72 +72 +48+72+72 +48 +48 +48 +32

>V _ 72
\)

8 8/ 512

Digitized byGoOgle

Original from
UNIVERSITY OF MICHIGAN

576 APPENDIX A

^ (continued)

r 1 =

32+40+8+40+8+16+32 + 16+48+8+16+8+48 + 16+32+16+8+

512

+40+8+40+32
512

64/3 2 3\ _ 192 + 128+192
64\8 + 8 + 8/ ~ 512

128 + 192 + 192

D-

512

64/3 3 2\ _ 192 + 192 + 128
64\8 8 + 8/ ~ 512

128+64 + 128+64 + 128

r =

512

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS

APPENDIX A

DISTRIBUTIVE INVOLUTION GROUPS

579

S80

APPENDIX A

582

APPENDIX A

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS

583

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION , GROUPS

585

586 APPENDIX A

F. Generalization: Distributive Cube of Polynomials (n Terms).

(1) Factorial: (a+b+c+.. . +m)» = [(a»+a l b+a , c+. . . + a l m) + ( a »b +
+ab*-fabc+.. . -fabm) + (a 2 c4-abc4-ac 2 4- • . . 4-acm) -+-,..+ (a 2 m+abm +
+acm + - . .+am 1 )] + [(a 2 b+ab l +abc + . . . +abm) 4-(ab 2 4-b 3 4-b 2 c4- • . .

. . .+b J m) + (abc+b J c+bc J +. . . +bcm) + . . . +(abm+b 2 m+bcm + . . .
. . .+bm 2 )] + [(a 2 c+abc+ac 2 +. . . +acm) + (abc4-b 2 c+bc 2 4- . . . +bcm) +
+ (ac l +bc 2 +c'+ . . . +c 2 m) 4- . . . + (acm 4-bcm +c 2 m 4- . ■ • 4-cm 2 )] + • . ■
... 4- [(a 2 m 4-abm 4-acm 4- • ■ ■ 4-am 2 ) 4- (abm 4-b 2 m 4-bcm 4- ■ • • 4-bm 2 ) +
4- (acm 4-bcrn +c 2 m 4- • • 4-cm 2 ) 4-- • • 4- (am 2 4-bm 2 4-cm 2 4- . . . 4-m 3 )];

(2) Fractional :( — 1 1 h

\a4-b4-c4- • . 4-m a4-b4-c4- ■■ . 4-m a4-b4-c4- • . • 4~ m

m \* if a 3 a 2 b

;)'= {[

a4-b+c4-. ■ - 4-m/ lL(a4-b+c4- . . . 4-m) 3 (a4-b4-c + . . . 4-m)'
a 2 c a 2 m | |~ a 2 b

1 ' * ' ' ( q _l_K _l_f _l_ _l_rr,^J |_(o O-K-L^-L -LmM - '"

(a-fb+c4-. . . 4-m) 1 (a+b+c+ . . . +m)»J L(a+b+c + . . . +m)»

ab 2 abc abm

(a4-b4-c-r-. • -4-m) 3 (a+b4-c + . . . 4-m)* (a+b+c-H . . . + m )

[a 2 c abc ac 2

(a+b4-c4-. • - 4-m)» + (a4-b4-c4-. . . 4-m) I+ (a4-b4-c4- . . . 4-m)» + '

4-

acm 1 I a 2 m abm

(a+b+c + . . .+m)»J L(a4-b4-c4-. . .4-m) 1 (a+b+c-t-. . . + m )»
acm am 2 ~|) (f a 2 b

(a+b+c + . . .+m)» (a+b+c-f- - - 4-m) 8 JJ (L(a4-b4-c4- . . . +m) 3

+

ab 2 abc abm

+ ■ ■ • ~r ', ; ; ; r; +

(a+b4-c4-. ■ • 4-m) 3 (a4-b+c4- . . - 4-m) 3 (a+b+c + . . . 4-m) 1

I" ab 2 b 3 bjc

+ L(a4-b4-c4-. ■ • 4-m) 3 + (a+b4-c4- . . 4-m) 3 + (a+b4-c4- . . . 4-m) 3 + '

b 2 m ~| f abc b 2 c

4-

.]+[

(a4-b4-c4-. . . 4-m) 3 J L(a4-b4-c4-. ■ • 4-m) 3 (a+b+c + . . . +m)»

be 2 bem I f abm

4-... 4 +■•■+ h

(a4-b+c4-. • • 4-m) 3 (a4-b4-c4-...4-m) 3 J L(a4-b4-c-^-...4-m) ,

b 2 m bem bm 2

+ r-r-. + - ■• + ,..■■ , r~ r, \t +

(a4-b4-c4-. . . 4-m) 3 (a4-b4-c4-. . ■ 4-m) 3 "' (a+b+c + . . . +m)

J}

Digitized by GoOglC

Original from
UNIVERSITY OF MICHIGAN

+

{[

DISTRIBUTIVE INVOLUTION GROUPS 587
a l c abc ac*

(a+b+c + . . . +m)» (a+b+c+. . . +m)» (a+b+c + . . . +m)»
acm 1 , f abc b'c

(a+b+c + . . .+m)»J L(a+b+c + ...+m)> (a+b+c + ...+m)«

be* bem "I . f ac*

, + ■••+•

(a+b+c + . . .+m)» (a+b+c+. . . +m)»J L(a+b+c + . . . +m)»

. +m)»] + [(

(a+b+c + . . .+m)» (a+b+c + . . . +m)» ' (a+b+c + . . . +m)

(

0-

acm bem c 2 m

.(a+b+c + . . . +m)» (a+b+c + . . . +m)» (a+b+c + . . . +m)»

■ cm * 11 ■ .if* a'm abm

' (a+b+c+...+m)»J)" r """ r lL(a+b+c+...+m)»" r (a+b+c+...+m)»

acm . ^ am

• ]+r — — +

. . . +m)'J L(a+b+c + . ..+m)>

(a+b+c + ...+m)» (a+b+c +

b'm . bem bm J

I * * * I i i \ * I *

(a+b+c + ...+m)» (a+b+c + ...+m)» " (a+b+c + . . . +m)'

tacm bem c*m

(a+b+c + . . .+m)« (a+b+c+. . . +m)» (a+b+c + . . . +m)» '

cm 1 1 f am 1 bm 2

' ' (a+b+c+...+m)»J ' ' ' ' L(a+b+c+...+m)» (a+b+c + . . . +m)»

+

cm 1 m'

+ ...+;

1 (a+b+c + ...+m)« " (a+b+c + . . . +m)

Synchronization of the first power group with the third power group.

(1) Factorial: S - a (a+b+c + . . . +m) l +b (a+b+c + . . .+m) l +

+c (a+b+c + . . . +m) J +. . . +m (a+b+c+. . . +m) J ;

p .• i c a / a+b+c + ...+m Y

(2) Fractional: S = — • ( — J +

a+b+c + ...+m \a+b+c + . - . + m/

b ^ / a+b+c-f ■ ■ -r-m V c

a+b+c + . . . +m " \a+b+c + . . . +m/ a+b+c+. . . +m '

/ a+b+c-K ._+m \* m / a+b+c+. - +m V

\a+b+c + . . . +m/ " a+b+c + . . . +m* \a+b+c+. . . +m/ '

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

588 APPENDIX A

Synchronization of the second power group with the third power group.

(1) Factorials = [a»(a+b+c + . . . +m)+ab (a+b+c + . . . +m) + *
+ac (a+b+c + . . .+m) + . . . +am (a+b+c + . . . +m)] +

+[ab (a+b+c+. . . +m)+b l (a+b+c + . . .+m)+bc (a+b+c + . . .+m)+. .

. . .+bm (a+b+c + . . .+m)]+[ac (a+b+c+ . . . +m) +

+bc (a+b+c + . . .+m)+c*(a+b+c + . . .+m) + . .'.

. . ,+cm (a+b+c + . . .+m)]+. . . +[am (a+b+c+. . .+m) +

+bm (a+b+c+. . . +m)+cm (a+b+c + . . . +m) + . . .

...+m , (a+b+c+...+m)];

(2) Fractional: S = , , — - • 1 ... , — J +

L(a+b+c + . . . +m)* \a+b+c + • • • +m/

ab /a+b+c + . . . +m\ ac

(a+b+c + . . .+m)« ' \a+b+c+. . . +m/ (a+b+c + . . . +m)» '

/ a+b+c + ...+m \ am / a+b+c + . . . +m V|

\a+b+c + . . .+m/ + - + (a+b+c+...+m) f "\a+b+c + ...+m/J

[ ab / a+b+c+...+m \ b'

L(a+b+c+. . • +m)« ' \a+b+c + . . . +m/ (a+b+c + . . . +m)» "

( a+b+c + . • • +m \ be /a+b+c + . . . +m\

a+b+c + . . +m/ (a+b+c+. . . +m) 1 ' \a+b+c + . . . +m/

, bm / a+b+c + . ■ ■ + m Y| , f ac ^

" (a+b+c + . . . +m)» ' \a+b+c+. . . +m/J L(a+b+c + . . . +m) J *

/ a+b+c+...+m \ be / a+b+c + . . . +m \

\a+b+c + . . . +m/ (a+b+c+ . . . +m)» ' \a+b+c+ . . . +m/

, cj / a+b+c + . . • +m\ cm ^

(a+b+c+. . .+m)« ' \a+b+c + . . . +m/ ' (a+b+c + . . . +m)» *

/ a+b+c + ...+m \*] f am / a+b+c+. ■ . +m \

\a+b+c + . . . +m/J ' ' L(a+b+c+- . • +m) J ' \a+b+c + . . . +m)

bm /a+b+c + . . . +m\

(a+b+c + . . .+m) 8 " \a+b+c+. . . +m)

( a+b+c + . . . +m \ m'
a+b+c+ . . . +m) * ' ' (a+b+c + . . ..+

cm

(a+b+c+. . .+m) J "

Digitized by GoOgle

+m) s " Va+b+c + . . . +m/J '

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS 589

+

(2) Fractional: S = • + ■

ac / a+b+c V] f ab / a+b+c \ b 2
(a+b+c) 2 ' \a+b+c/J L(a+b+c) 2 ' \a+b+c/ (a+b+c) 2 '

/ a+b+c \ be / a+b+c V| I" ac / a+b+c \

\a+b+c/ (a+b+c) 2 " \a+b+c/J L(a+b+c) 2 ' \a+b+c/

be / a+b+c \ c 2 / a+b+c V|

(a+b+c) 2 ' \a+b+c/ (a+b+c) 2 ' \a+b+c/J '

Synchronization of the first power group with the third power group.

(1) Factorial: S = a (a+b+c) 2 +b (a+b+c) 2 +c (a+'o+c) 2 ;

r . , c a /a+b+c\ 2 , b /a+b+cV ,

(2) Fractional : S = - ■ 1 : ) H • I I +

v ' a+b+c \a+b+c/ a+b+c \a+b+c/

/ a+b+c V
' \a+b+c/ '

c

+

a+b+c \a+b+c/

/. Cube of Quintinomials

6
6

+l)+(l+l+2 + l+l) + (2 +2+4+2+2) +

(\ ,1,2 1 IV (1+1+2 + 1
\6 + 6 + 6 + 6 + 67 =

216

+ (1+1 +2 + 1 +!)+(! +1+2 + 1 +!) + (! +1+2 + 1+1) +

216

+ (l+l+2 + l+l)+(2+2+4+2+2)+(l+l+2 + l+l) +

216

+ (l+l+2 + l+l)+(2+2+4+2+2)+(2+2+4+2+2) +

216

+ (4+4+8+4+4)+(2+2+4+2+2)+(2+2+4+2+2) +

216

+ (l+l+2 + l+l)+(l+l+2 + l+l) + (2+2+4+2+2) +

216

+(1+1+2+1 +l)+(l+l+2 + l+l)+(l +1+2 + 1+1) +

216

+(l+l+2+l+l)+(2+2+4+2+2)+(l+l+2 + l+l) +

216

+(1+1+2 + 1+1)

216

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

590 APPENDIX A

6

- (continued)
6

(l+2 + l+l+l)+(2+4+2+2+2)+(l+2+l+l+l) +

216

+(l+2+l+l+l)+(l +2+1+1 +l) + (2+4+2+2+2) +

216

+ (4+8+4+4+4)+(2+4+2+2+2)+(2+4+2+2+2) +

216

+(2+4+2+2+2)+(l+2 + l+l+l)+(2+4+2+2+2) +

216

+(l+2+l+l+l)+(l +2 + 1+1 +!)+(! +2 + 1 +1+1) +

216

+ (1 +2 + 1+1 +Q+(2+4+2+2+2)+(l +2 + 1+1+1) +

216

+(1+2 + 1+1 +!)+(! +2 + 1 +i+Q+(i +2+1+1+Q +

216

+ (2+4+2+2+2)+(l +2 + 1+1 +!)+(! +2+1+1+Q +

216

+ (1+2 + 1 + 1+1)
216

(2 11 11 V = (8+4+4+4+4) + (4+2+2+2+2)+(4+2+2+2+2) +
\6 6 6 6 6/ 216

+ (4+2+2+2 +2)+(4+2+2+2+2)+(4+2+2+2+2) +

216

+(2 + 1+1+1 +l)+(2+l+l+l+l) + (2 + l +1+1+Q +

216

+ (2 + l+l+l+l)+(4+2+2+2+2) + (2 + l +1+1+Q +

216

+(2+l+l+l+l)+(2 + l+l+l+l)+(2 + l+l+l+l) +

216

+ (4+2+2+2+2)+(2 + l+l+l+l) + (2 + l + l+l+Q +

216

+ (2 + l+l+l+l) + (2 + l+l+l+l)+(4+2+2+2+2) +

216

+(2+l+l+l+l)+(2+l+l+l+l) + (2+l +1+1+Q +

216

+(2+1+1+1+1)

216

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS

591

(continued)

6

(I 1 1 1 2\» = (1+1+1 + l+2)+(l+l + l+l+2)+(l +1+1 +1+2) +
\6 6 6 6 6/ 216

+(l+l+l+l+2)+(2+2+2+2+4) + (l+l+l+l+2) +

216

+(l+l+l+l+2)+(l+l+l+l+2)+(l+l+l+l+2) +

216

+ (2+2+2+2+4) +(1+1 + 1+1 +2) +(1+1 +1+1 +2) +

216

+(l+l+l+l+2)+(l+l+l+l+2)+(2+2+2+2+4) +

216

+(l+l+l+l+2) + (l+l+l+l+2)+(l+l+l+l+2) +

216

+(l+l+l+l+2)+(2+2+2+2+4) + (2+2+2+2+4) +

216

+(2+2+2+2+4)+(2+2+2+2+4) +(2+2+2+2+4) +

216

+(4+4+4+4+8)
216

/l 1 1 2 lV _ (1+1+1 +2 + l)+(l +1+1 +2 + l)+(l +1+1 +2 + 1) +
\6 6 6 6 6/ 216
■ +(2+2+2+4+2)+(l+l+l+2 + l)+(l + l +1+2 + 1) +

216

+(l+l+l+2 + l)+(l+l+l+2 + l)+(2+2+2+4+2) +

216

+ (l+l+l+2 + l)+(l+l+l+2 + l)+(l+l+l+2 + l) +

216

+ (l+l+l+2 + l)+(2+2+2+4+2)+(l+l+l+2 + l) +

216

+(2+2+2+4+2) +(2+2+2+4+2) + (2+2+2+4+2) +

216

+(4+4+4+8+4)+(2+2+2+4+2) +(1+1 +1+2 + 1) +

216

+(1+1+1 +2 + 1) +(l+l+l+2 + l)+(2+2+2+4+2) +

216

+(1+1+1+2 + 1)
216

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

592

APPENDIX A

6

{continued)

58(1) +2 + 1 +1+2+32(1) +2 + 20(1) +2+32(1) +2 + 1 +1+2+58(1)

+ (12 + 12+24 + 12 + 12) +(6+6 + 12 +6+6) +(6+6 + 12+6+6)

216

6/1 2 1 1 lV = (6 + 12+6+6+6)+(12+24 + 12 + 12 + 12) +
6\6 6 6 6 6/ 216

+ (6 + 12+6+6+6) +(6 + 12+6+6+6) +(6 + 12+6+6+6)

6/2 11 1 lV = (24 + 12 + 12 + 12 + 12) +(12+6+6+6+6) +
6\6 6 6 6 6/ 216

+ (12+6+6+6+6) +(12 +6+6+6+6) +(12+6+6+6+6)

216

6/1 11 1 2\» = (6+6+6+6 + 12) +(6+6+6+6 + 12) +
6\6 6 6 6 6/ 216

+ (6+6+6+6 + 12) +(6+6+6+6 + 12) +(12 + 12 + 12 + 12+24)

216

6/1 11 2 lV = (6+6+6 + 12+6) + (6+6+6 + 12+6) +
6\6 6 6 6 6/ 216

+ (6+6+6 + 12+6) + (12 + 12 + 12+24 + 12) +(6+6+6 + 12+6)

216

6\6 6 6 6 6/

(6+6 + 12+6+6) + (6+6 + 12+6+6) +
216

216

216

36(6)

216

36+36+36+36 + 72
" 216

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS

(concluded)

593

36/1 1 1 2 l\ _ 36+36+36+72+36
36\6 6 6 6 6/ ~ 216

_ 6 ( 36 )
f ~ 216

(\ , 2 , 1 2 IV (l+2 + l+2 + l) + (2+4+2+4+2)+( l+2 + l+2 + l) +
V7 + 7 + 7 + 7 + 7/ = 343

+ (2+4+2+4+2) +(1+2 + 1 +2 + 1) +(2+4+2+4+2) +

343

+ (4+8+4+8+4) +(2+4+2+4+2) +(4+8+4+8+4) +

343

+ (2+4+2+4 + 2) +(1+2 + 1 +2 + 1) +(2+4+2+4 + 2) +

343

+ (1+2 + 1 +2 + 1) +(2 +4+2+4+2) +.(1+2 + 1 +2 + 1) +

343

+ (2 +4+2+4 + 2) +(4+8+4+8+4) +(2+4+2+4 + 2) +

343

+ (4+8+4+8+4) +(2 +4+2+4+2) +(1+2 +1+2+1) +

343

+ (2+4 + 2+4 + 2) +(1+2 + 1 +2 + 1) +(2+4+2+4 + 2) +

343

+ (1+2 + 1+2 + 1)
343

(2 1 2 1 lV = (8+4+8+4+4) + (4+2+4+2+2) +(8+4+8+4+4) +
\7 7 7 7 7/ 343

+ (4 + 2+4 + 2+2) +(4+2+4 + 2+2) +(4 + 2+4+2 + 2) +

343

+ (2 + 1 +2 + 1+1) +(4+2+4+2+2) +(2 + 1 +2 + 1+1) +

343

+ (2 + 1 +2 + 1+1) +(8+4+8+4+4) + (4+2+4+2+2) +

343

+ (8+4+8+4+4) +(4 + 2+4 + 2+2) +(4 + 2+4+2 + 2) +

343

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

594

APPENDIX A

- (continued)

+ (4+2+4+2+2)+(2 + l +2 + 1 + 1) +(4+2+4+2+2) +

343

+(2 + 1 +2 + 1 +l)+(2 + l +2 + 1+1) +(4+2+4+2+2) +

343

+ (2 + 1 +2 + 1 +l)+(4+2+4+2+2) + (2 + l +2 + 1+1) +

343

+(2+1+2 + 1+1)
343

(I 2 1 1 2\" = (l+2 + l+l+2)+(2+4+2+2+4) + (l+2+l+l+2) +
\7 7 7 7 7/ 343

+ (l+2 + l+l+2)+(2+4+2+2+4) + (2+4+2+2+4) +

343

+ (4+8+4+4+8) +(2+4+2+2+4) + (2 +4+2+2+4) +

343

+ (4+8+4+4+8)+(l+2 + l+l+2)+(2+4+2+2+4) +

343

+ (1 +2 + 1 +l+2)+(l +2 + 1 +1+2) +(2 +4+2+2+4) +

343

+ (1+2+1 +1+2) +(2 +4+2+2 +4) +(1+2 + 1 +1+2) +

343

+ (1+2 + 1 +1+2) +(2+4+2+2+4) +(2 +4+2+2+4) +

343

+ (4+8+4+4+8)+(2+4 + 2+2+4) + (2+4+2+2+4) +

343

+ (4+8+4+4+8)
343

(2 1 1 2 lV

(-+;+-+-+-) =
\7 7 7 7 7/

(8+4+4+8+4)+(4+2+2+4+2)+(4+2+2+4+2) +

343

+ (8+4+4+8+4) +(4+2+2+4+2) +(4+2+2+4+2) +

343

+ (2 + l+l+2 + l) + (2 + l+l+2 + l)+(4+2+2+4+2) +

343

+ (2 + 1 +1+2 + 1) +(4 + 2+2+4+2) +(2 + 1 +1+2 + 1) +

343

Digitized byGoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS 595

- (continued)

+ (2 + 1 + l+2 + l)+(4+2+2+4+2)+(2 + l +1+2 + 1) +

343

+ (8+4+4+8+4)+(4+2+2+4+2)+(4+2+2+4+2) +

343

+ (8+4+4+8+4) + (4+2+2+4+2)+(4+2+2+4+2) +

343

+ (2 + 1 + 1 +2 + l)+(2 + l+l+2 + l)+(4+2+2+4+2) +

343

+(2 + 1+1+2+1)
343

/l 1 2 1 2\« = (1+1+2 + 1 +2) + (l+l +2+1 +2)+(2+2+4+2+4) +
\7 7 7 7 7/ 343

+ (l+l+2 + l+2)+(2+2+4+2+4) + (l + l+2 + l+2) +

343

+ (l+l+2 + l+2)+(2+2+4+2+4) + (l+l+2 + l+2) +

343 5
+ (2+2+4+2+4)+(2+2+4+2+4) +(2+2+4+2+4) +

343

+ (4+4+8+4+8) +(2+2+4 + 2+4) +(4+4+8+4+8) +

343

+ (l+l+2 + l+2) + (l+l+2 + l+2)+(2+2+4+2+4) +

343

+ (1+1 +2 + 1 +2) +(2+2+4+2+4) +(2+2+4 + 2+4) +

343

+ (2+2+4+2+4)+(4+4+8+4+8) + (2+2+4+2+4) +

343

+ (4+4+8+4+8)
343

r» =

9(l)+2 + ll(l)+2 + 19(l)+2 + l +2 + 1 +1+1 +2 + 15(1) +2 + 14(1) +

343

+2 + l+l+2+4(l)+2+27(l)+2 + l+3 + 7(l)+2 + 10(l)+2+6(l)+2 +

343

+27(l)+2+6(l)+2 + 10(l)+2 + 7(l)+3 + l+2+27(l)+2+4(l)+2 + l +

343

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

596 APPENDIX A

- (continued)

+l+2 + 14(l)+2 + 15(l)+2 + l+l+l+2+l+2 + 19(l)+2 + ll(l)+2 +

343

+9(1)
343

/l 1 3 1 lV = (1+1 +3 + 1 +!)+(!+! +3 + 1 +l)+(3+3+9+3+3) +
\7 7 + 7 7 7/ 343

+(l+l+3 + l+l)+(l+l+3 + l+l) + (l+l+3 + l+l) +

343

+(l+l+3+l+l)+(3+3+9+3+3)+(l +1+3 + 1+1) +

343

+ (l+l+3+l+l)+(3+3+9+3+3) + (3+3+9+3+3) +

343

+ (9+9+27+9+9)+(3+3+9+3+3)+(3+3+9+3+3) +

343

+(1+1 +3+1 +!)+(!+! +3 + 1 +l)+(3+3+9+3+3) +

343

+ (1+1 +3+1 +!) + (! +1+3 + 1 +!) + (! +1+3+1+1) +

343

+ (1+1 +3 + 1 +l) + (3+3+9+3+3)+(l+l +3 + 1+1) +

343

+(1+1+3 + 1+1)
343

(l 3 1 1 lV = (1 +3 + 1+1 +l)+(3+9+3+3+3)+(l+3+l+l+l) +
\7 + 7 + 7 7 7/ 343

+ (1+3+1 +!+!)+(! +3 + 1 +l+l)+(3+9+3+3+3) +

343

+(9+27+9+9+9)+(3+9+3+3+3)+(3+9+3+3+3) +

343

+ (3+9+3+3+3)+(l+3 + l+l+l)+(3+9+3+3+3) +

343

+(l+3 + l+l+l)+(l +3 + 1 +!+!)+(! +3 + 1 +1+1) +

343

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS

7 /

- (continued)

597

+(l+3 + l+l+l)+(3+9+3+3+3)+(l+3 + l+l+l) +

343

+(1+3 + 1 +1+1) +(1+3 + 1 +1+1) +(1+3 + 1+1+1) +

343

+(3+9+3+3+3) +(1 +3 + 1 +1+1)+(1 +3 + 1 +1+1) +

343

+ (1+3 + 1+1+1)
343

/3 1 1 1 lV = (27+9+9+9+9)+(9+3+3+3+3) +
\7 7 7 7 7/ 343

+ (9+3+3+3+3) + (9+3+3+3+3)+(9+3+3+3+3) +

343

+ (9+3+3+3+3) +(3 + 1 +l+l+l)+(3 + l +1+1+1) +

343

+ (3 + 1+1 +l+l)+(3 + l+l+l+l)+(9+3+3+3+3) +

343

+ (3 + 1 +l+l+l)+(3 + l +1+1+1) +(3 + 1 +1+1+1) +

343

+ (3 + 1 +1+1+1) +(9+3+3+3+3) +(3 + 1 +1 + 1+1) +

343

+ (3 + 1 +1+1+1) +(3 + 1 +l+l+l)+(3 + l +1+1+1) +

343

+ (9+3+3+3+3) +(3 + 1 +l+l+l) + (3 + l +1+1+1) +

343

+ (3 + 1 +l+l+l) + (3 + l +1+1+1)
343

/l 1 1 1 3\« = (l+l+l+l+3) + (l+l + l+l+3) + (l+l+l+l+3) +
\7 7 7 7 7/ 343

+ (1+1 +1 + 1 +3) +(3+3+3+3+9) +(1+1 +1+1 +3) +

343

+ (1+1 +1+1 +3) +(1+1 +1+1 +3) +(1+1 +1+1 +3) +

343

+ (3+3+3+3+9) + (1+1 +1+1 +3) +(1+1 +1+1 +3) +

343

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

598 APPENDIX A

7

(continued)

+ (1 + 1+1 +j +3)+(l +1 +1 + 1 +3)+(3+3+3+3+9) +

343

+ (1+1 +1+1 +3) +(1+1 +1+1 +3) + (1+1 +1+1 +3) +

343

+ (1+1 + 1+1 +3) +(3+3+3+3+9) +(3+3+3+3+9) +

343

+ (3+3+3+3+9) + (3+3+3+3+9) + (3+3+3+3+9) +

343

+ (9+9+9+9+27)
343

/l l 1 3 lV = (l+l+l+3 + l)+(l+l+l+3 + l) + (l+l+l+3 + l) +
\7 7 7 7 7/ 343

+ (3+3+3+9+3)+(l+l+l+3 + l)+(l+l+l+3 + l) +

343

+ (l+l+l+3 + l) + (l +1+1 +3 + 1) +(3+3+3+9+3) +

343

+ (l+l+l+3 + l)+(l +1+1 +3 + 1) +(1+1 +1+3 + 1) +

343

+ (l+l+l+3 + l) + (3+3+3+9+3) + (l+l+l+3 + l) +

343

+ (3+3+3+9+3) +(3+3+3+9+3) +(3+3+3+9+3 ) +

343

+(9+9+9+27+9)+(3+3+3+9+3) + (l+l+l+3 + l) +

343

, +(1+1+1 +3 + l)+(l+l+l+3 + l)+(3+3+3+9+3) +

343

+ (1+1+1+3 + 1)
343

. 25(l)+2 + 10(l)+2+28(l)+2 + 12(l)+2 + l+l+l+2+6(l)+2+6(l) +
r i _

343

+2+5(l)+2 + 15(l)+2+3+6(l)+2 + ll(l)+2+6(l)+2+5(l)+2 + 7(l) +

343

+2+5(l)+2+6(l)+2 + ll(l)+2+6(l)+3 + l+2 + 15(l)+2+5(l)+2 +

343

+6(l)+2+6(l)+2 + l+l+l+2 + 12(l)+2+28(l)+2 + 10(l)+2+25(l)

Digitized by GoOgle

343

Original from

UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS

7

- (continued)

Y 1 +?+i+?+ 1 Y = (7+14

7\7 7 7 7 7/

+7 + 14+7) +(14+28 + 14+28 + 14) +

343

+ (7 + 14+7 + 14+7) +(14+28+14+28 + 14) +(7 + 14+7 + 14+7)

7/2 12 1 lV (28 + 14

-\~+z+z+z+-) =-

7\7 7 7 7 7/

343

+28 + 14+14) +(14 + 7 + 14 + 7 + 7) +

343

+ (28 + 14+28 + 14+14) +(14+7 + 14 + 7 +7) + (14+7 + 14+7 +7)

7/12 11 2V (7 + 14

r(r+r+z+-+- 7 ) =

7\7 7 7 7 7/

343

+7+7 + 14) +(14+28 + 14 + 14+28) +

343

+ (7 + 14+7+7 + 14)+(7 + 14 + 7+7 + 14) +(14+28 + 14 + 14+28)

343

(28 + 14 + 14+28 + 14) +(14+7 + 7 + 14 + 7) +

343

+(14+7+7 + 14+7) + (28 + 14 + 14+28 + 14) +(14+7+7 + 14 + 7)

7/2 112 lV
7l7 + 7 + 7 + 7 + 7) "

343

(7+7 + 14+7 + 14) + (7+7 + 14 + 7 + 14) +

343

+ (14 + 14+28 + 14+28) +(7 +7 + 14 + 7 + 14) +(14 + 14+28 + 14+28)

343

_ 49(7)
343

(7+7+21 +7 + 7) +(7 + 7 +21 +7+7) +

7/113 1 lV

343

+ (21 +21 +63+21 +21) +(7 +7 +21 +7 + 7) +(7 + 7 +21 +7 +7)

343

+ 7 + 7)+(21 +63+21+21+21) +

7/1 3 1 1 A 2 = (7+21+7
7\7 7 7 7 7/

343

+ (7+21+7+7 + 7) + (7+21 +7 +7 +7) + (7 +21 +7+7 + 7)

7/3 111 lV
A7+7 + 7+7+7) "

343

(63+21 +21 +21 +21) +(21 +7 + 7 +7 +7) +

343

+ (21 +7+7+7+7) +(21 +7 + 7+7 +7) + (21 +7 + 7 +7+7)

343

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

600

7

- (concluded)

APPENDIX A

7/1111 3V

(7 +7 + 7 + 7+21) +(7+7+7+7+21) +
343

+ (7+7+7+7+21) +(7+7 + 7+7+21) + (21 +21 +21 +21 +63)

343

(7+7+7+21 +7) + (7 + 7+7+21 +7) +

343

+ (7 + 7 + 7 + 21+7)+(21+21+21+63+21) + (7 + 7+7+21+7)

343

r* =

49(7)
343

49+98+49+98+49
343

98+49+98+49+49
343

49/1 2 1 1 2\ _ 49+98+49+49+98
49\7 + 7 + 7 + 7 + 7/ ~ " 343

49/2 1 1 2 l\ _ 98+49+49+98+49
49\7 + 7 + 7 + 7 + 7/ ~ 343

49+49+98+49+98

49/1 2 1 , 2 , l\

±9^2 1 , 2 1 IN
4^V7 + 7 + 7 + 7 + 7 y

)/l 2 1 1 2\
)/2 1 1 2 l\

«(!+!+?+!+?)

49\7 7 7 7 7/

343

_ 7(49)
r 343

49 A ! 3 1 A 49+49 + 147+ 49+49
49\7 + 7 + 7 + 7 + 7/ " 343

49 + 147+49+49+49
343

49/3 1 1 1 1\ _ 147+49+49+49+ 49
49\7' f 7 + 7 + 7~ , ~7/ ~ 343

49+49+49+49 + 147
343

49+49+49 + 147+49

)/l 1 3 1 , l\

49\7 7 7 7 7/
)/3 1 1 1 . l\

49\7 7 7 7 7/

49/1 1 1 3 l\
4-9i7 + 7 + 7 + 7 + 7)

343

r 343

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS

601

8
8

(2 1 2 1 2\» = (8 +4 +8 -f 4 +8) +(4 +2 +4 +2 +4) +(8 +4 +8+4 +8) +
\8 8 8 8 8/ 512

+ (4 +2 +4 +2 +4) + (8 +4 +8 +4 +8) + (4 +2 +4 +2 +4) +

512

+ (2+l+2 + l+2)+(4+2+4+2+4)+(2 + l+2+l+2) +

512

+ (4 +2 +4 +2 +4) + (8 +4 +8 +4 +8) + (4 +2 +4 +2 +4) +

512

+(8+4+8+4+8)+(4+2+4+2+4)+(8+4+8+4+8) +

512

+(4+2+4+2+4)+(2 + l+2+l+2)+(4+2+4+2+4) +

512

+(2+l+2 + l+2) + (4+2+4+2+4)+(8+4+8+4+8) +

512

+ (4 +2 +4 +2 +4) + (8 +4 +8 +4 +8) + (4 +2 +4 +2 +4) +

512

+(8+4+8+4+8)
512

/l 2 1 2 2\« (l+2+l+2+2)+(2+4+2+4+4)+(l+2+l+2+2) +
V8 + 8' , "8 + 8 + 8/ = 512

+ (2 +4 +2 +4 +4) + (2 +4 +2 +4 +4) + (2 +4 +2 +4 +4) +

512

+(4+8+4+8+8) +(2 +4+2 +4+4) + (4+8+4+8+8) +

512

+ (4+8+4+8+8)+(l+2 + l+2+2)+(2+4+2+4+4) +

512

+(1 +2 + 1+2 +2) + (2+4+2+4+4) +(2 +4+2+4+4) +

512

+ (2 +4 +2 +4 +4) + (4 +8 +4 +8 +8) + (2 +4 +2 +4 +4) +

512

-[-(4-1-8+4+8+8) +(4+8+4+8+8) +(2 +4+2+4+4) +

512

+(4+8+4+8+8) + (2+4+2+4+4)+(4+8+4+8+8) +

512

+(4+8+4+8+8)
512

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

602

APPENDIX A

- (continued)

(2 1 2 2 lV = (8+4+8+8+4)+(4+2+4+4+2)+(8+4+8+8+4) +
\8 8 8 8 8/ 512

+(8+4+8+8+4)+(4+2+4+4+2) + (4+2+4+4+2) +

512

+(2+l+2+2 + l)+(4+2+4+4+2) + (4+2+4+4+2) +

512

+(2+l+2+2+l)+(8+4+8+8+4) + (4+2+4+4+2) +

512

+(8+4+8+8+4)+(6+4+8+8+4)+(4+2+4+4+2) +

512

+(8+4+8+8+4)+(4+2+4+4+2) + (8+4+8+8+4) +

512

+ (8+4+8+8+4)+(4+2+4+4+2) + (4+2+4+4+2) +

512

+ (2+l+2+2 + l)+(4+2+4+4+2)+(4+2+4+4+2) +

512

+(2+1+2+2 + 1)
512

/l 2 2 1 2\« _ (l+2+2+l+2) + (2+4+4+2+4)+(2 +4+4+2+4) +
\8 + 8 + 8 + 8 + 8/ 512

+ (1 +2+2 + 1 +2)+(2+4+4+2+4)+(2 +4+4+2+4) +

512

+(4+8+8+4+8)+(4+8+8+4+8)+(2+4+4+2+4) +

512

+(4+8+8+4+8)+(2+4+4+2+4) + (4+8+8+4+8) +

512

+ (4+8+8+4+8) + (2+4+4+2+4) + (4+8+8+4+8) +

512

+(1 +2+2+1 +2)+(2+4+4+2+4)+(2+4+4+2+4) +

512

+(1+2+2+1 +2) + (2 +4+4+2+4) +(2+4+4+2+4) +

512

+ (4+8+8+4+8)+(4+8+8+4+8) + (2+4+4+2+4) +

512

+(4+8+8+4+8)
512

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS 603

8

- (continued)

(2+2 1 2 IV = (8
\8 8 8 8 8/

+8 +4+8+4)+(8+8+4+8+4) + (4+4+2+4+2) +

r« =

512

+ (8+8+4+8+4)+(4+4+2+4+2) + (8+8+4+8+4) +

512

+ (8+8+4+8+4) + (4+4+2+4+2) +(8+8+4+8+4) +

512

+(4+4+2+4+2) +(4+4+2+4+2) + (4+4+2+4+2) +

512

+ (2+2 + 1 +2 + 1) +(4+4+2+4+2)+(2+2 + l +2 + 1) +

512

+(8+8+4+8+4) + (8+8+4+8+4) + (4+4+2+4+2) +

512

+ (8+8+4+8+4) + (4+4+2+4+2) + (4+4+2+4+2) +

512

+(4+4+2+4+2) + (2+2 + l+2 + l) + (4+4+2+4+2) +

512

+ (2+2 + 1+2 + 1)
512

l+2 + l+l + l+7(2)+4+4(l)+6(2)+6(l)+7(2)+4+8(2)+4+4(2) +

512

+5(4)+8(2)+4+7(2)+6(l)+9(2)+4(l)+6(2)+3(l)+2+5(l)+6(2) +

512

+4+4+1 +2 + 1+36(2) + ! +2 + 1 +4+4+6(2) +5(1) +2+3(1) +6(2) +

512

+4(l)+9(2)+6(l) + 7(2)+4+8(2)+5(4)+4(2)+4+8(2)+4+7(2) +

512

+6(l)+6(2)+4(l)+4+7(2) + l+l+l+2 + l

8/2 1 2 1 2\ 2 _
8\8 + 8 + 8 8 8/

512

(32 + 16+32 + 16+32) +(16+8 + 16+8 + 16) +

512

+ (32 + 16+32 + 16+32) + (16+8 + 16+8 + 16) +(32 + 16+32 + 16+32)

512

8/1 2 1 2 2V (8 + 16+8 + 16 + 16) + (16+32 + 16+32+32) +

'(i+H+H) ! - ~

8\8 8 8 8 8/ 512

+ (8 + 16+8 + 16 + 16) +(16+32 + 16+32+32) +(16+32 + 16+32+32)

512

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

604
8

(concluded)

(V+-+-+-Y =

\8 8 8 8 8/

APPENDIX A

(32 + 16+32 +32 + 16) + (16+8 + 16+16+8) +
512

+(32 + 16+32 +32 + 16) + (32 + 16+32 +32 + 16) + (16+8 + 16 + 16+8)

8\8 8 8 8 8/

512

(8 + 16 + 16+8 + 16)+(16+32 +32 + 16+32) +
512

+ (16+32+32 + 16+32) +(8+16 + 16+8 + 16) + (16+32+32 + 16+32)

512

8/2 2 1 2 lV _ (32 +32 + 16+32+16)+(32 +32+16+32 + 16) +
8\8 8 8 8 + 8/ ~ 512

+ (16 + 16+8 + 16+8) +(32+32+16+32 + 16) +(16 + 16+8 + 16+8)

512

r* =

8+16+8+8+8+6(16) + 10(8) +4(16) +10(8) +6(16) +8+8+8 + 16+8

64\8 8 8 8 8/

64\8 8 8 8 8/

64/2 12 2 l\

64\8 8 8 8 8/

64V8 8 8 8 8/

«(? +! A A +!)

64V8 8 8 8 8/

_ 8(64)
' " 512

512

128+64 + 128+64 + 128
512

64+128+64+128+128
512

128+64+128 + 128+64
512

64+128 + 128+64 + 128
512

128 + 128+64 + 128+64
512

Digit

Google

Original from
VERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS 605

9
9

/l 3 1 3 lV = (i+3+l+3+l)+(3+9+3+9+3)+(l+3 + l+3 + l) +
\9 9 9 9 9/ ~ 729

+ (3+9+3+9+3) + (l+3 + l+3 + l)+(3+9+3+9+3) +

729

+ (9+27+9+27 +9) +(3+9+3+9+3) +(9+27 +9+27 +9) +

729

+ (3+9+3+9+3) +(1+3 + 1 +3 + 1) +(3+9+3+9+3) +

729

+ (1 +3 + 1 +3 + l)+(3+9+3+9+3) +(1+3 + 1 +3 + 1) +

729

+ (3+9+3+9+3) + (9+27+9+27+9) + (3+9+3+9+3) +

729

+ (9+27+9+27+9) +(3+9+3+9+3) + (l +3 + 1 +3 + 1) +

729

+(3+9+3+9+3) +(1 +3 + 1 +3 + l)+(3+9+3+9+3) +

729

+ (1+3 + 1+3 + 1)
729

/3 1 3 1 lV = (27+9+27+9+9) + (9+3+9+3+3) +
\9 9 9 9 9/ " 729

+(27+9+27+9+9) + (9+3+9+3+3) + (9+3+9+3+3) +

729

+(9+3+9+3+3) + (3 + l +3 + 1+1) +(9+3+9+3+3) +

729

+(3 + l+3 + l+l)+(3 + l+3 + l+l)+(27+9+27+9+9) +

729

+ (9+3+9+3+3)+(27+9+27+9+9) + (9+3+9+3+3) +

729

+ (9+3+9+3+3) + (9+3+9+3+3) + (3 + l +3 + 1+1) +

729

+ (9+3+9+3+3) +(3 + 1 +3 + 1+1) +(3 + 1 +3 + 1+1) +

729

+ (9+3+9+3+3) +(3 + 1 +3 + 1 +l) + (9+3+9+3+3) +

729

+(3 + 1 +3 + 1 + 1) +(3 + 1 +3 + 1+1)
729

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

606 APPENDIX A

9

- (continued)

A 3 1 1 A* = (1 +3+1 + 1 +3) + (3+9+3+3+9) +(1+3+1 +1+3) +
\9 9 9 9 + 9/ 729

+(1 +3 + 1 +1 +3) +{3 +9+3 +3 +9) +<3 +9 +3 +3 +9) +

729

+(9+27+9+9+27) + (3+9+3+3+9) + (3+9+3 +3+9) +

729

+(9+27+9+9+27) + (l +3 + 1 + 1+3) +(3+9+3+3+9) +

729

+ (1 +3+1 +1 +3) + (l +3+1 +1 +3) + (3+9+3+3+9) +

729

+(1+3 + 1+1+3) +(3 +9 +3 +3 +9) +(1 +3 +1 +1 +3) +

729

+ (1+3+1 +1 +3) + (3+9+3 +3 +9) +(3 +9+3 +3 +9) +

729

+ (9 +27 +9 +9 +27 )+ (3 +9 +3 +3 +9) +(3 +9+3+3 +9) +

729 ,

+(9+27+9+9+27)
729

/3 1 1 3 l\* _
\9 + 9 + 9 + 9 + 9/

(27+9+9+27 +9) +(9+3 +3 +9+3) +
729

+(9+3+3+9+3)+(27+9+9+27+9) + (9+3+3+9+3) +

729

+ (9+3+3+9+3)+(3+l+l+3+l)+(3 + l +1+3 + 1) +

729

+ (9+3+3+9+3)+(3+l + l+3+l)+(9+3+3+9+3) +

729

+(3 + 1 +1+3+1) +(3 + 1 +l+3+l)+(9+3 +3+9+3) +

729

+(3 + l + l+3+l)+(27+9+9+27+9) + (9+3+3+9+3) +

729

+(9+3+3+9+3) + (27+9+9+27+9) + (9+3+3+9+3) +

729

+ (9+3+3+9+3) + (3+l +1+3 + 1) +(3+1 +1+3+1) +

729

+ (9+3+3+9 + 3)+(3 + l +1+3 + 1)
729

Cnr\n\(> Original from

UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS 607

9

- {continued)

(l 13 1 3V = (l+l+3 + l+3)+(l+l+3 + l+3) +
\9 9 9 9 9/ 729

+(3+3+9+3+9)+(l +1+3 + 1 +3) +(3+3+9+3+9) +

729

+ (1+1+3 + 1 +3) + (l +1+3 + 1 +3) + (3+3+9+3+9) +

729

+(l+l+3 + l+3)+(3+3+9+3+9) + (3+3+9+3+9) +

729

+ (3+3+9+3+9) + (9+9+27+9+27) + (3+3+9+3+9) +

729

+ (9+9+27+9+27) + (l+l+3 + l+3)+(l+l+3 + l+3) +

729

+(3+3+9+3+9) + (l+l+3 + l+3) + (3+3+9+3+9) +

729

+ (3+3+9+3+9) +(3+3+9+3+9) + (9+9+27+9+27) +

729

+3+3+9+3+9) + (9+9+27+9+27)
729

. 1 + 1+2 + 1+1+2 + 1+1+1+1+2 + 1+ 3+3+3+3+6+3 + 1+3 + 1 +
r* =

729

+ 1 +3 + 7(1)4-7(3) +1+3 + 1 +3 + 1 +1+1 +1+2 + 1 +3 + 1 +1+1 +2 + 1 +

729

+4(3)+6+4(3) + l+l+3 + l+4(3)+6+3+9+3(3)+6+3+9 + 10(3)+6 +

729

+6(3)+9+7(3) + l +1+2 + 1 +1+3 + 1 +1+3 + 1 +3+6+4(3) +1+2 + 1 +

729

+ l+l+3 + 7(l)+3 + l +1+2 + 1 +7(3) + 1+1 +1+2 + 1 +1+2 + 1 +1+1 +

729

+2 + 1 +1+2 + 1 +1+1+7(3) + ! +2 + 1 +l+3+7(l)+3 + l +1+1 +2 +

729

+ l+4(3)+6+3 + l +3 + 1 +1+3 + 1+1 +2 + 1 +1 +7(3) +9+6(3) +6 +

729

+ 10(3) +9+3+6+3+3+3+9+3+6+4(3) + 1+3 + 1 +1+4(3) +6+4(3) +

729

+ 1+2 + 1 +1 + 1 +3 + 1 +2 + 1 +1+1 +1+3 + 1 +3 + 1 +7(3)+7(l) +

729

+2 + 1+1 +3 + 1 +3+6+4(3) + l+2+4(l) +2 + 1 +1+2 + 1+1

729

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

608

APPENDIX A

- (continue!

(2 2 1 2 2\* _ (8+8+4+8 +8) + (8 +8 +4+8 +8) +

+ (4+4+2 +4+4) + (8+8+4+8+8)+(8+8+4+8+8) +

729

+ (8+8+4+8+8)+{8+8+4+8+8) + (4+4+2+4+4) +

720

+ (8+8+4+8+8) +(8+8+4+8+8) + (4+4+2+4+4) +

729

+(4+4+2 +4+4) +(2+2 + 1 +2+2) + (4+4+2+4+4) +

729

+(4+4+2+4+4) +(8+8 +4+8+8) + (8+8+4+8+8) +

729

+ (4 +4 +2 +4 +4) +(8 +8 + 4 +8 +8) + (8 +8 +4 +8 +8) +

729

+ (8+8+4+8+8)+(8+8+4+8+8)+(4+4+2+4+4) +

729

+ (8+8+4+8+8)+(8+8+4+8+8)
729

ft I 2 2 2V _ (8+4+8+8+8) +(4+2+4+4+4) +
\9 + 9 + 9 + 9 9/ " 729

+(8+4+8+8+8) +(8+4+8+8+8) +(8+4+8+8+8) +

729

+ (4+2+4+4+4) +(2 + 1 +2+2+2)+(4+2+4+4+4) +

729

+(4+2+4+4+4)+{4+2+4+4+4)+(8+4+8+8+8) +

729

+(4+2+4+4+4) +(8+4+8+8+8) + (8+4+8+8+8) +

729

+(8+4+8+8+8)+(8+4+8+8+8) + (4+2+4+4+4) +

729

+(8+4+8+8+8) + (8+4+8+8+8) +(8+4+8+8+8) +

729

+(8+4+8+8+8) + (4+2+4+4+4) +(8+4+8+8+8) +

729

+ (8+4+8+8+8) + (8+4+8+8+8)
729

Google

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS 609

(continued)

1 2 2 2 2\» = (1+2+2+2+2) +(2 +4+4+4+4) +
9 9 9 9 9/ 729

+ (2 +4+4+4+4) +(2 +4+4+4+4) + (2 +4+4+4+4) +

729

+ (2 +4+4+4+4) + (4+8+8+8+8) +(4+8+8+8+8) +

729

+ (4+8+8+8+8) + (4+8+8+8+8) +(2+4+4+4+4) +

729

+ (4+8+8+8+8) + (4+8+8+8+8) + (4+8+8+8+8) +

729

+ (4+8+8+8+8) + (2+4+4+4+4)+(4+8+8+8+8) +

729

+ (4+8+8+8+8) +(4+8+8+8+8) + (4+8+8+8+8) +

729

+ (2 +4+4+4+4) + (4+8+8+8+8) +(4+8+8+8+8) +

729

+ (4+8+8+8+8) + (4+8+8+8+8)
729

2 2 2 2 1 V _ (8+8+8+8+4) +(8+8+8+8+4) +
9 + 9 + 9~ , ~9 + 9/ ~ 729

+ (8+8+8+8+4) + (8+8+8+8+4) +(4+4+4+4+2) +

729

+ (8+8+8+8+4) + (8+8+8+8+4)+(8+8+8+8+4) +

729

+ (8+8+8+8+4) +(4+4+4+4+2)+(8+8+8+8+4) +

729

+ (8+8+8+8+4) + (8+8+8+8+4)+(8+8+8+8+4) +

729

+ (4+4+4+4+2) + (8+8+8+8+4) + (8+8+8+8+4) +

729

+ (8+8+8+8+4) +(8+8+8+8+4) +(4+4+4+4+2) +

729

+ (4+4+4+4+2) + (4+4+4+4+2) +(4+4+4+4+2) +

729

+ (4+4+4+4+2)+(2+2+2+2 + l)
729

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

610 APPENDIX A

9

- (continued)

(2 2 2 1 2\» = (8+8+8+4+8)+(8+8+8+4+8) +
\9 9 9 9 9/ 729

+(8+8+8+4+8) +(4+4+4+2 +4) + (8+8+8+4+8) +

729

+ (8+ 8 +8+4+8) +(8+8+8+4+8) +(8+8+8+4+8) +

729

+ (4+4+4+2+4) + (8+8+8+4+8) + (8+8+8+4+8) +

729

+ (8+8+8+4+8) +(8+8+8+4+8) +(4+4+4+2+4) +

729

+ (8+8+8+4+8) + (4+4+4+2+4) + (4+4+4+2+4) +

729

+ (4+4+4+2+4) + (2+2+2 + l+2) + (4+4+4+2+4) +

729

+ (8+8+8+4+8) + (8+8+8+4+8) +(8+8+8+4+8) +

729

+(4+4+4+2+4) + (8+8+8+4+8)
729

, 1+2+ 2+2+1+1+2 + 1+3+1+3 + 1+3 + 1+3+1+1+3 + 1+3+1 +
r* = ■

729

+3 + 1 +1+2 + 1 +1 + 1 +3 + 1 + 1+2 + 1+1 +3 + 1 +2+4(l)+2 + l +3 +

729

+ 1+1+2 + 1+3 + 1+1+1+3 + 1+1+2 + 1+4 + 1+2 + 1+1+2 + 1+1 +

+2+2+2 + 1+1+4+2 + 1+1+2+2+2 + 1+5+2+1+3 + 1+1+2+2 +

729

+2+1+1+2+2+2 + 1+1+4+2 + 1+3+2+2 + 1+1+3 + 1+2+2 +

729

+2 + 1+2+1+1+2 + 1+1+2 + 1+1+3 + 1+4+3 + 1+4+3+1+4+2 +

729

+ 1+3+1+1+2 + 1+3 + 1+1+2 + 1+1+3+1+3 + 1+3 + 1+1+1+3 +

729

+5(l)+2+2 + l+2 + l+l + l+2 + l+2+2+4(l)+4+8(l)+2 + l+4 +

729

+ 1+2 + 1+4+1+2 + 1+1+2 + 1+1+2+2+2+1+1+4+2 + 1+3+4 +

729

rv in _j i» r\rscs\f> Original from

Digitized by ^UU^IC UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS 611

9

- (continued)

+1+1+3+1+2 + 1+1+3 + 1+2+2+2 + 1+3+4 + 1+1+2 + 1+2 + 1 +

729

729

+ 1+1+2+2+2 + 1+1+2 + 1 + 1+2 + 1+4 + 1+2 + 1+4 + 1+2+8(1) +

729

+4+4(l)+2+2 + l +2 + 1 +1+1 +2 + 1 +2+2+5(1) +3 + 1 +1+1 +3 +

729

729

729

729

729

729

+1+3 + 1+3 + 1+3 + 1+2 + 1+1+2+2+2 + 1
729

?\9 9 9 9 9/

(9+27+9+27 +9) + (27 +81 +27 +81 +27) +
9\9 ' 9 " 9 ' 9 ' 9/ 729

+ (9+27+9+27 +9) + (27 +81 +27+81 +27) +(9+27 +9+27 +9)

_ ^ _

(81 +27 +81 +27 + 27) +(27 +9+27 +9+9) +

9/3 13 1 lV

729

+ (81 +27 +81 +27+27) + (27 +9+27 +9+9) +(27 +9+27 +9+9)

729

(9 +27 +9+9+27) + (27 +81 +27+27+81) +
9\9 ' 9 ' 9 " 9 ' 9/ 729

+ (9+27+9+9+27) +(9+27+9+9+27) +(27+81 +27+27+81)

729

j/l 3 1 1 3V

Digitized b/ Google

612 APPENDIX A

9

(continued)

9

9/3 1 1 3 lV (81+27+27+81 +27)+(27+9+9+27+9) +

J/3 113 lV
?\9 9 + 9 + 9 + 9/

9\9 9 9 9 9/ 729

+ (27 +9+9+27 +9) +(81 +27 +27 +81 +27) +(27 +9+9+27 +9)

729

9/113 1 3V (9+9+27+9+27)+(9+9+27+9+27) +

)/l 1 3 1 3V

9\9 9 9 9 9/ 729

+ (27 +27+81 +27 +81) + (9+9+27 +9+27) + (27 + 27 +81 +27 +81)

729

. 9+9 + 18+9+9 + 18+4(9) + 18+9+27 +27 +27 +27 +27 + 7(9) +
r s — .

729

+ 18+9+9+9 + 18+7(9) +27+27 +27 +27+27 +9 + 18+4(9) + 18+9 +

729

+9 + 18+9+9
729

(36+36 + 18+36+36) + (36+36 + 18+36+36) +

9/2 2 12 2\ s

-,(,+,+,+,+,; -

729

+ (18 + 18+9 + 18 + 18) +(36+36 + 18+36+36) + (36+36 + 18+36+36)

729

9/2 , 1 , 2 2 ,2V (36 + 18+36+36+36) + (18+9 + 18 + 18 + 18) +

J/2 1 2 2 2\» =
J\9 9 9 9 9/

9\9 9 9 9 9/ 729

+ (36 + 18+36+36+36) + (36 + 18+36+36+36) + (36 + 18+36+36+36)

9/1 2 2 2 2V _
9\9 + 9 9 + 9 9/

729

(9+18 + 18+18 + 18) + (18+36+36+36+36) +

729

+ (18+36+36+36+36) +(18+36+36+36+36) +(18+36+36+36+36)

9/2 2 2 2 lV
- 9 ( 9 + 9 + 9 + 9 + 9 ) -

729

(36+36+36+36 + 18) +(36+36+36+36 + 18) +

729

+ (36+36+36+36 + 18) +(36+36+36+36 + 18) + (18 + 18 + 18+18+9)

729

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS 613

9

^ (concluded)

9/2 2 2 1 2V (36+36+36 + 18+36)+(36+36+36+18+36) +

J/2 2 2 1 2V

;(,+,+,+,+,) -

9\9 9 9 9 9/ 729

+ (36+36+36 + 18+36) + (18 + 18 + 18+9 + 18) +(36+36+36+18+36)

729

9 + 18+9(9) + 18+9+9 + 18+5(9) + 18+9(9) + 18+9+9+18+9 + 18 +

2 _

r

729

+9+9 + 18+9(9) + 18+5(9) + 18+9+9+18+9(9) + 18+9

729

81/1 3 1 3 l\ _ 81+243+81+243+81
8lV9 + 9 + 9 + 9 + 9/ ~ 729

81/3 1 3 1 l\ _ 243+81+243+81+8 1
81\9 9 9 9 9/ ~ 729

81/1 3 1 1 3\ _ 81+243+81+81+243
81\9 + 9 + 9 + 9 9/ ~ 729

•0

1 3\
9 + 9)

81/3 1 1 3 l\ _ 243+81+81+243+8 1
81\9 9 9 9 9/ ~ 729

81/1 1 3 1 3\ _ 81+81+243+81+24 3
81\9 9 9 9 9/ ~ 729

9(81)

r =

729

81/2 2 1 2 2\ _ 162 + 162+81+162+162
8l\9 + 9 + 9 + 9 + 9/ ~ 729

D

1/2 1 2 2 2\
l\9 + 9 + 9 + 9 9/

D

81V9 9 9 9 9/

81/2 1 2 2 2\ _ 162+81+162 + 162 + 162

81\9 + 9 + 9 + 9 + 9/ " 729

81/1 2 2 2 2\ _ 81+162 + 162 + 162 + 16 2

81\9 + 9 + 9 + 9 + 9/ _ 729

81/2 ,2 2 2 l\ _ 162 + 162 + 162 + 162+81

81\9 9 9 9 9/ ~ 729

162 + 162 + 162+81+162

729

9(81)
729

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

616

APPENDIX A

DISTRIBUTIVE INVOLUTION GROUPS

617

616

APPENDIX A

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

618

APPENDIX A

DISTRIBUTIVE INVOLUTION GROUPS

APPENDIX A

Digitized by GoOgle

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

622 APPENDIX A

DISTRIBUTIVE INVOLUTION GROUPS

623

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

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Digitized by ^.UUglC UNIVERSITY OF MICHIGAN

DISTRIBUTIVE

INVOLUTION

GROUPS

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626

APPENDIX A

DISTRIBUTIVE INVOLUTION GROUPS

627

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

DISTRIBUTIVE INVOLUTION GROUPS

629

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

630

APPENDIX A

DISTRIBUTIVE INVOLUTION GROUPS

631

632

APPENDIX A

DISTRIBUTIVE INVOLUTION GROUPS

633

634

APPENDIX A

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

•

636

APPENDIX A

777. GROUPS OF VARIABLE VELOCITY.

These groups represent various forms of acceleration and growth.

Each group can be used in positive and in negative form (direction).

Both directions can be combined simultaneously and/or in sequence.

The resultant of interference of the same group in simultaneity of both di-
rections (the resultant of acceleration) may be used as an additional component.

The resultant of acceleration has an axis of symmetry, and always consists of
an odd number of terms, regardless of the origin of its generator.

The rate of acceleration in some of these groups is constant (as in the geometric
progressions or the involution series) and in some — variable (as in the natural
Integers or the summation series).

The presentation of variable velocities in the form of series containing only
integers makes it possible to solve many important problems in the practical
execution of design and music.

A. Rhythms of Variable Velocities

1. Natural Harmonic Series

1, 2,3,4,5,6, 7,8,9
r = 1+2+3+3+1+5+2+4+3+4+2+5+1+3+3+2+1

2. Arithmetical Progressions

1, 3, 5, 7, 9, 11, 13, 15, 17, 19
r - 1+3+5+7+3+5 + 11+13+2 + 13+11+5+3+7+5+3+1

1. 4, 7, 10, 13, 16, 19, 22
r = 1+4+7+10 + 13+6+10+6 + 13 + 10+7+4+1

J. Geometrical Progressions

r =

1,2,4, 8, 16,32

1+2+4+8+16 + 1+16+8+4+2 + 1

r —

1,3,9,27

1+3+9 + 14+9+3+1

r =

3,6, 12,24,48

3+6+12+24+3+24+12+6+3

r =

2,6,18,54

2+6 + 18+28 + 18+6+2

Digitize

Original from
UNIVERSITY OF MICHIGAN

GROUPS OF VARIABLE VELOCITY

637

4. Power Series

2,4,8, 16,32
r = 2+4+8 + 16+2 + 16+8+4+2

3,9,27,81
r = 3+9+27+42+27+9+3

5. Summation Series

1,2,3,5,8,13,21
r = i+3+5+8+4+9+4+8+5+3+l

1,3,4,7,11,18
r = 1+3+4+7+3+8+3+7+4+3+1

1,4,5,9, 14, 23
r = i+4+5+9+4+10+4+9+5+4+l

6. Arithmetical Progressions with Variable Differences

1,2,4,7,1.1,16, 22,29
r = 1+2+4+7 + 11+4 + 12+10+12+4+11+7+4+2+1

7. Prime Number Series

1,2,3,5,7, 11, 13, 17, 19, 23
r = 1+2+3+5+7+5+6 + 12+17+13+6+5+7+5+3+2+1

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UNIVERSITY OF MICHIGAN

638

APPENDIX A

GROUPS OF VARIABLE VELOCITY

639

638

APPENDIX A

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640

APPENDIX A

GROUPS OF VARIABLE VELOCITY

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UNIVERSITY OF MICHIGAN

APPENDIX B.
RELATIVE DIMENSIONS 1

'The reader is referred to Part II, Chapter 2,
Continuity, Section E, "Ratios of the Rational
Continuum," which offers a graphic presentation
of the ratios producing these relative dimensions.
(Ed.)

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Ratios of the rational continuum.

Digitized byGoOgle

Original from
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/. Relative Dimensions

240,000

16,000

32,000

160,000

15,484

30,968

120,000

240,000

15,000

30,000

96,000

192,000

14,545

29,090

80,000

160,000

14,118

28,236

68,571

137,142

13,714

27,428

60,000

120,000

13,333

26,666

53 333

106,666

12,973

25,946

48,000

96,000

12,632

25,264

43 636

87,272

12,308

24,616

40,000

80,000

12,000

24,000

36,923

73,846

11,707

23,414

34,286

68,572

11,429

23,258

32 000

64,000

11,163

22,326

30,000

60,000

10,900

21,800

28,235

56,470

10,667

21,334

26,667

53,334

10,435

20,870

25,263

50,526

10,213

20,426

24,000

48,000

10,000

20,000

22,857

45,714

21,818

43,636

20,870

41,740

20,000

40,000

19,200

38,400

18,462

36,924

17,778

35,556

17,143

34,286

645

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

646

APPENDIX B

1,00000
.66666
.50000
.40000
.33333
.28572
.25000
.22222
.20000
.18181
.16666
.15384
.14284
.13333
.12500
.11764
.11111
.10526

20. .10000

21. .09523

2,

3.

4.

5.

&

7.

8.

9,
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.

22.
23.
24.

25.

.09090
.08695
.08333
,08000

.33334
.16666
.10000
.06667
.04761
.03572
.02778
.02222
-.01819
.01515
.01282
.01100
.00951
.00833
.00736
.00653
.00585
.00526
.00477
.00433
.00395
.00362
.00333

26
27
28
20
30
31
32
33
34
35
36

17
38
3<>
40
41
42
43
44
45
46.
47
48

.07692
.07406
.07142

.06896
.06666
.06451
.06250
.06060
.05882
.05714
.05555
.05405
.05263
.05128
.05000
.04878
.04761
.04651
.04545
.04444
.04347
.04255
.04166

.00308

.00286

.00264

.00246

.00230

.00215

.00201

.00190

.00178

.00168

.159*

.150

.142

.135

.128

.122

.117

.110

.106

.101

.97

.92

.89

* The figures in this column represent the differ- be. In some cases one lero is omitted, and in

ence between the bracketed numbers to the left. others, two or three leros. By subtracting any

For the sake of visual simplicity, zeros are omit- bracketed figure from the one directly above it,

ted after the decimal point. The full figure would the reader may quickly determine how many zeros

be .00159 or .00097 or .01150, as the case may have been omitted. (Ed.)

Google

Original from
UNIVERSITY OF MICHIGAN

RELATIVE DIMENSIONS

647

4. 1.00000

5. .80000

6. .66666

7. .57144

8. .50000

9. .44444

10. .40000

11. .36362

12. .33332

13. .30768

14. .28568

15. .26666

16. .25000

17. .23528

18. .22222

19. .21052

20. .20000

21. .19046

22. .18180

23. .17390

24. .16666

25. .16000

26. .15384

.20000

.13334

.9522

.7144

.5556

.4444

.3638

.3030

.2564

.2200

.1902

.1666

.1472

.1306

.1170

.1052

.954

.866

.790

.724

.666

.616

27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.

.14812
.14284
.13792
.13332

.12902

]

.12500
.12120
.11764
.11428
.11110
.10810
.10526
.10256
.10000
.09756
.09522
.09302
.09090
.08888

]

.08694

]

.08510

]

.08332

.572
.528
.492
.460
.430
.402
.380
.356
.336
.318
.300
.284
.270
.256
.244
.234
.220
.212
.202
.194
.184
.178

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

648

APPENDIX B

6. 1.00000

7. .85716

8. .75000

9. .66666

10. .60000

11. .54543

12. .49998

13. .46152

14. .42852

15. .39999

16. .37500

17. .35292

18. .33333

19. .31578

20. .30000

21. .28569

22. .27270

23. .26085

24. .24999

25. .24000

26. .23076

27. .22218

.14284

.10716

.8334

.6666

.5457

.4545

.3846

.3300

.2853

.2499

.2208

.1959

.1755

.1578

.1431

.1299

.1185

.1086

.999

.924

.858

28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.

.21426
.20688
.19998
.19353
.18750
.18180
.17646
.17142
.16665
.16215
.15789
.15384
.15000
.14634
.14283
.13953
.13635
.13332
.13041
.12765
.12498

.792
.738
.690
.645
.603
.570
.534
.504
.477
.450
.426
.405
.384
.366
.351
.330
.318
.303
.291
.276
.267

See footnote on page 646.

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

RELATIVE DIMENSIONS

649

8. 1.00000

9. .88888

10. .80000

11. .72724

12. .66664

13. .61536

14. .57136

15. .53332

16. .50000

17. .47056

18. .44444

19. .42104

20. .40000

21. .38092

22. .36360

23. .34780

24. .33332

25. .32000

26. .30768

27. .29624

28. .28568

.11112

.8888

.7276

.6060

.5128

.4400

.3804

.3332

.2944

.2612

.2340

.2104

.1908

.1732

.1580

.1448

.1332

.1232

.1144

.1056

29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.

.27584
.26664
.25804
.25000
.24240
.23528
.22856
.22220
.21620
.21052
.20512
.20000
.19512
.19044
.18604
.18180
.17776
.17020
.17020
.16664

.984
.920
.860
.804
.760
.702
.672
.636
.600
.568
.540
.512
.488
.468
.440
.424
.404
.388
.368
.356

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

650

APPENDIX B

10. 1.00000

11. .90905

12. .83330

13. .76920

14. .71420

15. .66665

16. .62500

17. .58820

18. .55555

19. .52630

20. .50000

21. .47615

22. .45450

23. .43475

24. .41665

25. .40000

26. .38460

27. .37030

28. .35710

29. .34480

.9095
.7575
.6410
.5500
.4755
-4165
.3680
.3265
.2925
.2630
.2385
.2165
.1975
.1810
.1665
.1540
.1430
.1320
.1230

30.
31.
32.
33.
34.
35.
36.
37.
38.
39,
40.
41.
42.
43.
44.
45.
46.
47.
48.

.33^0
.32255
.31250
.30300
.29410
.28570
.27775
.27025
.26315
.25640
.25000
.24390
.23805
.23255
.22725
.22220
.21735
.21275
.20830

See footnote on page 646.

Google

Original from
UNIVERSITY OF MICHIGAN

•

RELATIVE DIMENSIONS

651

12. 1.00000

13. .92304

14. .85704

15. .79998

16. .75000

17. .70584

18. .66666

19. .63156

20. .60000

21. .57138

22. .54540

23. .52170

24. .49998

25. .48000

26. .46152

27. .44436

28. .42852

29. .41376

30. .39996

.7696
.6600
.5706
.4998
.4416
.3918
.3510
.3156
.2862
.2598
.2510
.2172
.1998
.1848
.1716
.1584
.1476
.1380

31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.

.38706
.37500
.36360
.35333
.34284
.33330
.32430
.31578
.30768
.30000
.29268
.28566
.27906
.27270
.26664
.26082
.25530
.24996

.1290

.1206

.1140

.1068

.1008

.954

.900

.852

.810

.768

.732

.702

.660

.636

.606

.582

.552

.534

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

652

APPENDIX B

14. 1,00000

15. .93331

16. .87500

17. .82348

18. .77777

19. .73682

20. .70000

21. .66661

22. .63630

23. .60865

24. .58331

25. .56000

26. .53844

27. .51842

28. .49994

29. .48272

30. .46662

31. .45157

.6669
.5831
.5152
.4571
.4095
.3682
.3339
.3031
.2765
.2554
.2331
.2156
.2002
.1848
.1722
.1610
.1505

32.

33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.

.43750
,42420
.41174
.39998
.38885
.37835
.36841
.35896
.35000
.34146
.33327
.32527
.31815
.31108
.30429
.29785
.29162

See footnote on page 646

RELATIVE DIMENSIONS

653

16. 1.00000

17. .94112

18. .88888

19. .84208

20. .80000

21. .76184

22. .72720

23. .69560

24. .66664

25. .64000

26. .61536

27. .59248

28. .57136

29. .55168

30. .53328

31. .51608

32. .50000

.5888
.5224
.4680
.4208
.3816
.3464
.3160
.2896
.2664
.2464
.2288
.2112
.1968
.1840
.1720
.1608

33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.

.48480
.47056
.45712
.44440
.43240
.42104
.41024
.40000
.39024
.38088
.37208
.36360
.35552
.34776
.34040
.33328

.1520

.1424

.1344

.1272

.1200

.1136

.1080

.1024

.976

.936

.880

.848

.808

.776

.736

.712

Digit

Google

Original from
UNIVERSITY OF MICHIGAN

654

18. 1.00000

19. .94734

20. .90000

21. .85707

22. .81810

23. .78255

24. .74997

25. .72000

APPENDIX B

33

26
27
28
29
30
31
32

.69228
.66654
.64278
.62064
.59994
.58059
.56250

.5266
.4734
.4293
.3897
.3555
.3258
.2997
.2772
.2574
.2376
.2214
.2070
.1935
.1809
.1710

34
35
36
37
38
39
40
41
42
43
44
45
46
47
48

.54540

.52938
.51426
.49995
.48645
.47367
.46152
.45000
.43902
.42849
.41859
.40905
.39996
.39123
.38295
.37494

.1602

.1512

.1431

.1350

.1278

.1215

.1152

.1098

.1053

.990

.954

.909

.873

.828

.801

Sec footnote on page 646.

Google

Origi
UNIVERSIT,

RELATIVE DIMENSIONS

655

20. 1.00000

21. .95230

22. .90900

23. .86950

24. .83330

25. .80000

26. .76920

27. .74060

28. .71420

29. .68960

30. .66660

31. .64510

32. .62500

33. .60600

34. .58820

.4770
.4330
.3950
.3620
.3330
.3080
.2860
.2640
.2460
.2300
.2150
.2010
.1900
.1780

35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.

.57140
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.45450
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.1680

.1590

.1500

.1420

.1350

.1280

.1220

.1170

.1100

.1060

.1010

.970

.920

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Original from
UNIVERSITY OF MICHIGAN

656

APPENDIX B

22. 1.00000

23. .95645

24. .91663

25. .88000

26. .84612

27. .81466

28. .78562

29. .75856

30. .73326

31. .70961

32. - .68755

33. .66660

34. .64702

35. .62854

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.3663
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36.
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Original from
UNIVERSITY OF MICHIGAN

RELATIVE DIMENSIONS

657

24. 1.00000

25. .96000

26. .92304

27. .88872

28. .85704

29. .82752

30. .79992

31. .77412

32. .75000

33. .72720

34. .70584

35. .68568

36. .66660

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Original from
UNIVERSITY OF MICHIGAN

6S8

APPENDIX B

26. 1.00000

27. .96278

28. .92846

29. .89648

30. .86658

31. .83863

32. .81250

33. .78780

34. .76466

35. .74282

36. .72215

37. .70265

28. 1.00000

29. .96544

30. 93324

31. .90314

32. .8750

33. .84840

34. .82348

35. .79996

36. .77770

37. .75670

38. .73682

See footnote on page 646.

.3722
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38
39
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42
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44
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39.

40.

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43

44

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.63630
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Original from
UNIVERSITY OF MICHIGAN

30. 1.00000

31. .96765

32. .93750

33. .90900

34. .88230

35. .85710

36. .83325

37. .81075

38. .78945

39. .76920

40. .75000

41. .73170

42. .71415

43. .69765

44. .68175

45. .66660

46. .65205

47. .63825

48. .62490

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.2250
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RELATIVE DIMENSIONS

32. 1.00000

33. .96960

659

34. .94112

35. .91425

36. .88880

37. .86480

38. .84208

39. .82048

40. .80000

41. .78048

42. .76176

43. .74416

44. .72720

45. .71104

46. .69552

47. .68080

48. .66656

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Original from
UNIVERSITY OF MICHIGAN

660

34
35
36
37
38
39
40

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.91885
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41. .82926

42. .80937

43. .79067

44. .77265

45. .74448

46. .73899

47. .72335

48. .70822

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.2550
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APPENDIX B

36
37
38
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41
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43. .83718

44. .81810

45. .79992

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47. .76590

48. .74988

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38. 1.00000

39. .97432

40. .95000

41. .92672

42. .90459

43. .88369

44. .86355

45. .84436

46. .82593

47. .80845

48. .79154

40. 1.00000

41. .97560

42. .95220

43. .93020

44. .90900

45. .88880

46. .86940

47. .85100

48. .83320

RELATIVE DIMENSIONS

42. 1.00000

661

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660

APPENDIX B

34. 1.00000

35. .97138

36. .94435

37. .91885

38. .89471

39. .87176

40. .85000

41. .82926

42. .80937

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44. .77265

45. .74448

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39. .92304

40. .90000

41. .87804

42. .86698

43. .83718

44. .81810

45. .79992

46. .78246

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UNIVERSITY OF MICHIGAN

APPENDIX C.

NEW ART FORMS

I. Double Equal Temperament
II. Rhythmicon

III. SOLIDRAMA

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UNIVERSITY OF MICHIGAN

/. Double Equal Temperament 1

Double Equal Temperament is a primary selective system of tuning, designed
in accordance with this theory. Double Equal Temperament successfully unifies
all systems of intonation used in the western world of today. It consists of the
basic intervals: y/2, combined with micro-intervals: 'v^2, which serve as devia-
tion units and are arranged bifoldedly in relation to each basic unit. The micro~
units are best averages for all differences between the units of the y/2 and just
intonation (natural scale). This tuning permits one to execute with a high degree
of precision: twelve-unit equal temperament, mean temperament, just intona-
tion, and the string and vocal inflections of special types of intonation (chamber,
jazz, Gypsy music, etc.).

An electronic organ with micro-tuning and a specially designed keyboard was
built in 1932 for the author by Leon Theremin for the performance of Double
Equal Temperament.

II. Rhythmicon*

The Rhythmicon is an instrument constructed by Leon Theremin. It is the
first modern instrument that composes music automatically. The present model
is confined to the composition and automatic performance of rhythmic patterns
in the acoustical scale of intonation. The forms of rhythmic groups produced by
this instrument are the resultants of interference of generators from one to six-
teen.

The author found a special use for the Rhythmicon: reproduction of the most
intricate forms of aboriginal African drumming. Many phonograms have been
made to illustrate this, by means of eliminating the middle and the low fre-
quencies in both the recording and the performance.

The drum sounds obtained from the Rhythmicon are fully realistic, and the
configurations reach the intricacy of completely saturated sets.

The total number of resultants is 65,535.

The Rhythmicon is in the possession of The Schillinger Estate.

'The reader is referred back to Chapter 1,
"Selective Systems" of Part III, Technology of
Art Production. Double Equal Temperament
illustrates a primary selective system of tuning de-
vised by Schillinger to accommodate intonations

not possible in our present system of tuning. (Ed.)

*Thc Rhythmicon, in Schillinger's terminology,
produces secondary selective systems. The reader
is referred back to Chapter 1 , "Systems" of Part
III. Technology of Art Production. (Ed.)

665

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666

APPENDIX C

A. The Number of Forms Produced by the Rhythmicon

(C = combinations )

16

16!

16!

1

(16-

D!

1!15!

16!

16!

2

!(16-

2)!

2!14!

16!

16!

3

(16-

3)!

~ 3!13!

16!

16!

4

(16-

4)!

4!12!

16!

16!

5

(16—

5)!

5 ! 1 1 !

16!

16!

6

(16-

6)!

6!10!

16!

16!

7

!{16-

?)!

7!9!

16!

16!

8

(16-

8)!

8!8!

16!

16!

9

1(16-

9)!

9!7!

16!

16!

10

!(16-

10)!

10!6!

16!

16!

11

(16-

11)!

11 !5!

16!

16!

12

(16-

12)!

~ 12!4!

16!

16!

13

(16-

13)!

13!3!

16!

16!

14

(16-

14)!

14!2!

16!

16!

15

(16-

15)!

" isri s

16!

16

(16-

16)!

= 120
= 560

- 1820
= 4368

- 8008
= 11440
= 12870
= 11440
= 8008
= 4368

- 1820

- 560
= 120

- 16
= 1

TOTAL = 65535

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UNIVERSITY OF MICHIGAN

NEW ART FORMS

667

It would take 10,922.5 hours to play all the combinations if we gave each
combination the average duration of 10 seconds.

10,922.5 hours = 455 days, 2 hours, 30 minutes.

The eighteenth art form, 1 representing motion and transformation of solids.
Its components are: time and the three spatial coordinates.

Motion of screens and solids is executed by means of a magnetic drive in the
working model designed by the author. The name of the instrument is Solidrive. 1
This instrument gives a simultaneous performance of space-time configurations
in an automatic form.

The Solidrive may be synchronized with light and sound. The present model
has a 45-inch diameter and four symmetrically arranged drives. The Solidrive
is in the possession of The Schillinger Estate.

A. Forms of Solidrama

(1) Motion of solids: through their own trajectories or through a built-in-trajecto-
form.

(2) Motion of planes: vertical, horizontal, curved.

(3) Use of solids and planes as platforms, stages, etc.

(4) Use of solids and planes as screens for illumination (light projection [colors]).

(5) Use of solids and planes as luminescent or partly luminescent forms.

(6) Use of solids and planes as shadow-casters.

(7) Use of all the previous devices combined with mirror reflexions and multi-
reflexions.

(8) All the previous forms combined with diffusing screens.

(9) Motion of solids and planes combined with music.

(10) Combined arts: solids and planes, light (colors) and music (recitation).

777. Solidrama

•See the table in Chapter 1 of Part II.
•The Solidrive is protected by patent.

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UNIVERSITY OF MICHIGAN

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APPENDIX D.
POETRY AND PROSE 1

'In 1934 Schillinger lectured on the application
of his theory to poetry and prose before the
Faculty Club of Columbia University (Mathe-
matics Dept.)- The title of the lecture was
"Poetry and Prose Mathematically Devised."

Schillinger had planned to include a chapter on
the subject in Part III of the present work. The
material in this Appendix represents the begin-
ning of such a chapter. (Ed.)

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670

APPENDIX D

A. Elements of Poetic Structure

Different literary forms require different degrees of precision.
The structural unit of poetry may be:

(1) a syllable;

(2) a word;

(3) a sentence;

(4) a stanza; etc.

Poetic structures may be arranged into sonic and semantic scales. Sonic
scales represent: syllabic configurations, with specified accents, assonance and
alliteration. Semantic scales represent: direct and indirect (metaphoric) asso-
ciational classifications, and are arranged through the degrees of connotational
intensity (associative power).

This theory abolishes the duality of meter and rhythm, unifying both into
temporal structures.

B. Rhythmic Composition of Sentences

Each plot is subdivided into a uniform scale of events (episodes). The impor-
tance of event defines temporal stress. Thus events are rhythmically arranged,
each episode being expressed through a different number of sentences.

Application of the technique of expansion through growth of the determinant
of a series makes it possible to extend a short story into a novel.

Original from
UNIVERSITY OF MICHIGAN

APPENDIX E.
PROJECTS 1

I. Books
II. Instruments

'To present the application of his basic ideas
to all the arts, Schillinger planned a series of
books and instruments. His sudden death in 1943
prevented the completion of a project as fabulous

as it is fraught with the most significant implica-
tions for the future of the arts. Here, in typical
rhythmic form, is a graphic presentation of the
project "Books" as Schillinger set it down. (Ed.)

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672

APPENDIX E

PROJECTS

673

//. Instruments 1

Mechanical realization of this method is a natural consequence. Various
instruments may be constructed for the automatic production, reproduction,
and variation of works of art. Instruments of the analyzer type may also be
constructed for the automatic testing of the esthetic quality of works of art.
The following types of instruments are planned by the author in the form of
engineering design and kinetic diagrams:

1. Instruments for production, variation and reproduction of industrial design'
This group of instruments serves the purpose of automatic composition of design
in the following forms:

a. drawing

b. projection

c. printing

d. weaving

2. Instruments for automatic variation of design, using design in the following
forms:

a. drawing

b. films or slides

c. fabrics

3. Instruments for automatic composition of music:

a. limited to specified components, such as rhythm, melody,
harmony, harmonization of melodies, counterpoint, etc.

b. combining the above functions, and capable of composing an
entire piece with variable tone qualities (choral, instrumental
chamber music, symphonic and other orchestral music)

4. Instruments for automatic variation of music of the following types:

a. quantitative reproductions and variations of existing music

b. modernizing old music

c. antiquating modern music

5. Instruments of groups 3 and 4, combined with sound production for the
purpose of performance during the process of composition or variation.

6. Semi-automatic instruments for composing music. These instruments will
be used as a hobby for everyone interested in musical composition, whether
amateur or professional, and will not require any special training. The prospective
name for instruments of this type will be "Musamaton." 2 These instruments

'Plans for the construction of various instru-
ments are in the possession of Mrs. Joseph Schil-
linger. The right to construct these instruments
is protected by patent, and no instrument may
be constructed or used for private profit without
the written consent of The Schillinger Estate.

•These words, coined by Schillinger to describe
various instruments, have been registered, and
may not be used in connection with these or other
instruments without the written permission of
The Schillinger Estate.

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674

APPENDIX E

may be used for the purpose of entertainment and study, and will be suitable
for schools, clubs, public amusement places, and homes.

7. Instruments of the type described in paragraph 6, but in the field of design.
The prospective general name for such instruments will be "Artomaton." 2 The
two fundamental types of Artomatons will be:

a. "graphomaton"* — an instrument producing linear design

b. "luminaton"' — an instrument producing design projected by light source

8. Projecting optical instruments with the mechanism for automatic composi-
tion of form by color, and capable of projecting the latter during the process of
composing.

9. Instruments for kinetic displays, which may be used in exhibitions, per-
manent exhibits, stores, store windows.

10. Instruments for kinetic theatrical productions (as conceived through this
theory) including kinetic stage, light, sound, scent, taste, and tactile effects.

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Original from
UNIVERSITY OF MICHIGAN

GLOSSARY

Compiled by ARNOLD SHAW

(This glossary is limited to terms that have a special significance
and that are basic in Schillinger's system of thought. Mathematical
terms as such arc not included. Terms that appear within a defini-
tion in bold face arc explained in their alphabetical order. The
index at the close of this volume indicates the pages in the text
where the term occurs.)

a or A Denotes Attack

® Denotes the original position in Quadrant Rotation.

a -f- b Signifies the Resultant of Interference between a and b. The division sign, as used by
Schillinger, does not mean a divided by b, or a : b. It means Interference, not division,
and may be algebraically represented as follows: a -r b = b + (a — b) + (2a — 2b) +

+ (2a — 2b) + (a — b) + a. Arithmetically 4-8-3 = 1-. In interference 4 -5- 3 =

3+1+2 + 2 + 1+3 12

— 12 = » +b ' r »+b. and T,+b are other variant symbols of inter-
ference.

a -5- b Symbol for the Resultant of another type 1 of interference between a and b known as
Interference with Fractioning.

ABSCISSA ROTATION. A technique for varying the pitch or spatial pattern of an art work.
We begin with a graph on a plane. If we rotate the graph around its horizontal coordinate,
or abscissa, we secure a cylinder in horizontal position on which the lowest and highest parts
of the score or design meet.

ACCELERATION SERIES. Any series in which successive terms are the result of an increasing

differential, e.g., 1®, 2®, 4®, 7®, 11®, 16 When the differential decreases, the series may

be known as a Retardation Series. Arithmetical and Geometrical Progressions, the
various Power and Summation Series, Natural Harmonic Series, Prime Number
Series, etc., are sometimes also known as Acceleration Series.

AMPLITUDE. The loudness or intensity of a sound is measured by the amplitude, or greatest
displacement, of its air vibrations. In the graph of a sound wave, the amplitude is the dis-
tance between the time axis and the highest point of the wave.

ARITHMETICAL PROGRESSION. A series formed by the addition of a constant number to
each successive term of the series: e.g., 1, 4, 7, 10, 13, 16. . . . in which the constant is 3. A
slightly different form of this progression is encountered in the Acceleration Series 1, 2, 4,

7, 11, 16, 22, 29 Here the number being added is 1, then 2, then 3; 4; S; 6; etc. In this

form of progression, the constant is the difference (1) between the successive terms in the
additive group. This series is known as an Arithmetical Progression with Variable Difference.

ARTISTIC SCALE. See Operand Group.

ATTACK. In music, any tonal event. "Four attacks per measure" means four musical events
in each measure. These events may be four single tones, four chords, or four string attacks
without reference to rhythmic pattern.

ATTACK-GROUP. Simply a series of attacks considered as a unit.

AXIS OF SYMMETRY. See Symmetry.

675

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UNIVERSITY OF MICHIGAN

676

GLOSSARY

B Symbol for balance.

© Denotes the backward position in Quadrant Rotation.

B I - COORD I NATE ROTATION. Rotation through abscissa and ordinate. We begin with a
graph, or bi- coordinate system, on a plane. By rotating the graph around the vertical co-
ordinate, or ordinate, (tf» O or <t> O in reference to time) we obtain a cylinder in vertical
position. Rotating the graph around the abscissa, or horizontal coordinate, (4> O or O
in reference to pitch or density) produces a cylinder in horizontal position. In the first in-
stance, beginning and terminal durations meet. In the second case, when the vertical co-
ordinate represents pitch, the highest and the lowest pitches meet. In music, bi-coordinate
rotation is a technique for varying the textural density of the composition.

BINOMIAL. An algebraic expression consisting of two elements, e. g., a + b. In general, any
group composed of two elements,

© Denotes the backward and upside down position in Quadrant Rotation.

CIRCULAR PERMUTATION. A type of Permutation in which the original group or series
is re-arranged one step at a time. Such re-arrangement may proceed in a clockwise C' or
counter-clockwise O direction. 1, 2, 3, in clockwise circular permutation, becomes 2, 3, I and
3, 1, 2 before returning to 1, 2, 3. In counter-clockwise direction, the circular permutations
are 1, 3, 2; 3,2, 1; and 2, 1, 3. Circular permutation is also known as Displacement or Circu-
lar Displacement.

COEFFICIENTS OF RECURRENCE. When the Resultant* of Interference are used as a
form of regularity, they are known as coefficients of recurrence. Such coefficients serve to
control the periodicity of a given element in an art form — the number of times a color recurs
in a rhythmic design or the number of times a given interval or duration recurs in a musical
composition. Any rhythmic number series may serve as a coefficient of recurrence. Schil-
ling's procedure is to begin with a Primary Selective System, the color spectrum or the
tuning system. When coefficients of recurrence are applied to this system, we obtain a Color
Scale or a pitch scale. The further application of coefficients of recurrence to this Secondary
Selective System produces a color scheme or a melody.

COLOR SCALE. A sequence of colors produced by applying some rhythmic pattern or form of
regularity to the color spectrum. Such a sequence involves increasing or decreasing wave-
lengths, increasing or decreasing densities.

COMMON PRODUCT. In the process of deriving Resultants, a number obtained by multi-
plying the Generators.

COMPLEMENTARY FACTOR. The number of times a given Generator recurs in the process
of deriving Resultants of Interference. This is calculated by dividing the given generator
into the Common Product of all the generators involved.

CONFIGURATION. A pattern or rhythmic form. Schillinger uses configuration scale as another
term for Secondary Selective System.

CONTINUITY. A continuity is a finite portion of a Continuum. It is apparent because the
particular points either belong to an axiomatic class (e.g., the natural integers) or are part
of a harmonic group (e.g., Summation Series). In terms of a graph, a continuity is formed
by selecting rhythmic points along any coordinate or Parameter. In terras of an art form,
a continuity is an ordered sequence of elements.

CONTINUUM. A system of unlimited Parameters, or measuring lines. In music, the total
manifold of all possible frequencies. In design, the total manifold of points, lines, arcs, colors,
etc.

CONTRACTION GROUP. Any complex rhythmic group in which a longer duration group is
followed by a shorter duration group — both groups being derived from the same Style

Diqili

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677

Series. The longer group is generally the resultant of interference with fractioning (ry+ij)
while the shorter group is the resultant of simple interference (r, +b )' An Expansion Group
consists of the same groups, with the shorter preceding the longer.

COORDINATE CONTRACTION. See Coordinate Expansion.

COORDINATE EXPANSION. May be performed with reference either to the abscissa, thereby
affecting the general component, time, or with reference to the ordinate, expressing some
special art component. The process is one of multiplying the given component by a coeffi-
cient of 2, 3, 4 or more. In music, coordinate expansion is distinguished from tonal expansion
in that the simple semitone units of the graph serve as the basis in the former, and some se-
lected diatonic scale provides the basis in the latter. The reverse process of coordinate ex-
pansion is coordinate contraction, which cannot generally be achieved in music with our
present tuning system. In the visual arts, coordinate expansion produces forms of optical
aberration, elongation or distension such as we find in El Greco and other artists.

CORRELATION. Schillinger deals with three main types of correlation: parallel when two
series (quantities, directions or phases) increase at the same rate; contrary when one series
increases in value as the other decreases; oblique when one series remains constant while
the other increases or decreases. These correlations apply to the motion of two voices in
music, to the correspondence between pitch and time ratios, to the relation of density and
time in the composition of density groups, and to the correlation of any two components
in design.

COSINE CURVE. See Sine Curve.

D

<3) The fourth position in Quadrant Rotation — forward and upside down.

A, 8 The Greek letter Delta. Symbols for Density. A = compound density-group. A * =
sequent compound density group.

D, d Symbol used in composition of Density, d = density unit. D = simultaneous density-
group and D * = sequent density-group.

d Denotes zero displacement in Circular Permutation. The zero displacement is the initial
position and no displacement at all. di, dj, ds denote successive displacements.

DENSE SET. If we take a straight line, itself finite, we can insert an infinite number of points
or number values. All these points or numbers constitute a dense set. Viewed more generally,
a dense set consists of all the number values, both rational and irrational, in the space-time
Continuum. Primary and Secondary Selective Systems, which are developed on the
basis of uniform symmetric ratios or non-uniform rhythmic forms, are derived from the con-
tinuum.

DENSITY. The quantity of sound per unit of time in music. In design, the number of lines,
areas, arcs, colors, etc., per unit of space. In general, the criterion of judgment is the approach
to a Dense Set. In music, Density Groups vary according to whether they involve the
use of all available pitches and parts. In design, they vary as they involve all possible con-
figurations and colors. Density groups are subject to Phasic Rotation, symbolized by the
Greek letter </>. Such rotation may involve both coordinates and produce intercom position
of their phases; this process is symbolized by the Greek letter 0.

DERIVATIVE SCALE. Any scale developed by a process of General or Circular Permu-
tation from an original or Parent Scale.

DETERMINANT. The original value in a Style Series. — ..t...

t D t 3 tj t t

. . . t J .... t J .... t n ... ± ... I ... I 3 .... 9 .... 27 ... . -, or ;in this

27 9 3 3 t 3

instance, is the determinant.

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GLOSSARY

DEVIATION SERIES. A series involving forms of expansion and contraction, and produced
by adding a unit of growth (r) to a constant unit (t). A term in such a series would be repre-

I

sented as t = t' + t. The simplest relation of t' to t is f = ~, The numerator in this

relationship may grow through any of the Acceleration Series, such as Arithmetical and
Geometric Progressions, etc,

DEVIATION, STANDARD. A balanced binomial may be thrown out of balance through the
use of a standard deviation unit. This unit is added to one term of the binomial and sub-
tracted from the other. Formula for such a unit is t (unit of growth) = — (constant unit),

DISPLACEMENT. A method of permutating the elements in a group by consecutively re-
arranging their order, generally in one direction. For example:

do = a + b + c + . . . . + m
di = b + c + , + m + a
dt = c + . , , + m + a + b

Such re-arrangement may be clockwise O (forward) or counterclockwise O (backward).
Also known as Circular Permutation or mechanical permutation as distinguished from
General Permutation. The initial group is known as do, denoting zero or no displace-
ment, di is the first displacement, dj the second, etc.

DISTRIBUTIVE CUBE. See Distributive Involution.

DISTRIBUTIVE INVOLUTION. A process of raising a binomial or polynomial to any power
and arranging the product into its summary parts— in short, power differentiation. The
binomial a + b squared is a 3 + 2ab + b 2 . The distributive square, however, is aa + ab +
+ ba -|- bb. The non-distributive square of 4 + 3 is 49. Distributively the square is 16 +
+ 12 + 12 + 9. Schillinger found that the distributive use of powers was extremely valuable
in design and music, where they are occasionally used as Coefficients of Recurrence.

DISTRIBUTIVE POWERS. See Distributive Involution.

DISTRIBUTIVE SQUARE, See Distributive Involution.

DISTRIBUTIVE USE. See Distributive Involution.

DURATION. The time within which a sound lasts. Represented on a graph by extension along
the abscissa or horizontal coordinate. Any duration may be used as a unit in developing a
Continuity or as a phase in Positional Rotation.

DURATION GROUP. A group of one or more durations used as a pattern for rhythm in musk.

E

Eo Symbol for a Fitch-Scale in zero expansion, meaning that the scale cannot be contracted
in the given tuning system. Ei, Ej, etc., indicate the first Expansion, second Expansion, etc.

EQUAL TEMPERAMENT. The present system of tuning, which serves as the basis of most
Occidental music. This tuning system, like all Primary Selective Systems, is composed
of a particular set of pitches selected from the manifold, or Dense Set, of all possible pitches.
Equal temperament, developed by Andreas Werckmeister in 1691, involves the division of
the octave into 12 pitches whose frequencies are related to each other as the logarithmic
series based on the twelfth root of 2 (V». C = 2^ or 1, C# = 2'\ D = 2", D# = 2<\
E - 2* F = 2<\ F# - 2<\ G = 2", G# = |A, A = 2*. A# «2*i B = 2^,
C = 2« or 2. The frequency of each octave is twice the original tone, and remaining pitches
are derived by octave duplication. The frequency of concert A today is 440 vibrations per
second,

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679

EXPANSION. A process applied particularly to Pitch-Scales by which the successive pitches
are increased by some constant factor. When the results are determined tonally (i.e., diatonic-
ally) we have a Tonal Expansion. When they are determined exactly, we have a Geo-
metrical Projection or expansion. See Coordinate Expansion. Applicable also to du-
rations in music, and to quantities, extensions, densities, etc., in design.

EXPANSION GROUP. See Contraction Group.

F

F denotes factorial continuity while f denotes fractional continuity.

FACTORIAL. Refers to the organization of an art form as a whole while fractional concerns
individual units. The rhythmic structure of the whole is a factorial problem whereas the
rhythmic organization within a unit, a bar in music or a unit-area in design, is fractional.

FACTORIAL-FRACTIONAL CONTINUITY. A progression developed by inserting terms
between the existing terms in a Normal Series. The values at the left side of such a pro-
gression determine the fractional continuity (unit rhythms). The values on the right side

2 t

control the Factorial continuity (work as a whole). With the determinant - = -, the

2 t

following factorial-fractional continuity may be developed : — .... - ....

16 8 4 2

- 2 4 8 16

2

FAMILY-SERIES. See Style-Series.
FIBONNACI SERIES. See Summation Series.

FRACTIONING. The process of dividing a rhythmic group into fragments, generally on the

basis of polynomials in the Style Series. Also the specific process of producing resultants

known as Interference with Fractioning. The Resultant of 3 4-2 = 2+1 +1+2.

The resultant of 3-i-2 with fractioning = 2+1+1+1+1+1+2.
FREQUENCY. The number of vibrations per second of a vibrating medium. The frequency of

vibrations for middle C is 256 per second, while the frequency of C one octave higher is

512 vibrations per second.
FUNDAMENTAL TONE. The tone produced by the vibrations of the whole string or column

of air, as distinguished from overtones or Partlals produced by vibrations of portions of

the string or air column.

G

GENERAL PERMUTATION. See Permutation.

GENERATOR. A series of numbers (generally composing a uniform group) used in combination
with another uniform series to produce a new non-uniform group. The new group ,is known
as the Resultant, and the process, as Interference. The numbers may be converted into
durations, pitches, etc., in music and extensions, angles, colors, etc., in design.

GENETIC FACTOR. Schillinger regards phasic or periodic differences as the genetic factor in
art. If we take two uniform groups of durations with such differences and synchronize them,
Interference occurs. The Resultant is a non-uniform group, which may become the basis
of general or specific art components.

GEOMETRICAL PROGRESSION. Various number series formed by multiplying each succes-
sive term by a constant number: e.g., 1, 3, 9, 27, 81 The multiplier is 3. In 3, 6, 12,

24, 48, 96. . . .the multiplier is 2. Power Series frequently have the appearance of geomet-
rical progressions, but they are formed by a process of raising the initial term to its different
powers: e.g., 3, 9, 27, 81 ... ., which is evolved through 3 1 , 3 2 , 3 3 , 3 4

GEOMETRICAL PROJECTION. The general technique for varying art forms. Includes 1)
Quadrant Rotation, 2) Coordinate Expansion or Geometric Expansion, and 3) Co-
ordinate Contraction.

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li

HARMONIC GROUP. A group whose numbers, durations, extensions, etc., display Harmonic
Relations or a perceptible rhythmic regularity.

HARMONICS, In acoustics, the subcomponents of a sound wave, accessory to the fundamental
tone. Also known as Partlala and produced by physical factors that transform a simple
sound wave (Sine ware) into one of more complex form. See Natural Harmonic Series.

HARMONIC RELATIONS. Refers to the rhythm, design, or underlying regularity of a work
of art.

HIGHER ORDER. An operation of any type performed on the results of a previous operation
of the same type: e.g., squaring a square, cubing a cube, or grouping a group. Schillinger also
employs a variation technique that he calls Permutations of a Higher Order. If we desig-
nate a and b as our original elements, a i and bi become elements of the first order. au= ai +
4" bj and bs = bi + ai, are elements of a second or higher order, as = aj + bj and ba =
bj 4- ai is a further permutation of a higher order, etc., until a„ — a„. i -f- b n _ i and
b„ = b„_ i + a„- i.

HYBRID SERIES. A series that involves a mixture of types. The Determinants of a uniform

. . . ,. 2.3.4. n.

series are associated with the natural integer sets — - series, - series, - series .... - series.

2 3 4 n

When the members of a family belong to one series, it is considered pure. When the members

belong to several series, it is considered hybrid.

INTERFERENCE. One of the basic concepts of Schillinger's approach to the arts, interference
is a phenomenon observed in all fields of wave motion. When two sound waves or two light
waves of varying Periodicity cross, they interfere or combine to form a third wave that is
the summation of the two. Schillinger regarded this phenomenon as a process of growth and
evolution. For him, it became the fundamental procedure for combining two or more uni-
form periodicities to produce a new non-uniform group. This procedure, which he calls inter-
ference, is the foundation stone of the Theory of Regularity and Coordination in the pres-
ent work, and of the Theory of Rhythm in the Schillinger System of Musical Composition.

INVARIANT OF INVERSION. Denotes the axis or element around which an inversion or
Geometrical Projection is performed.

INVERSION. See Geometrical Projection and Quadrant Rotation.

INVOLUTION. See Distributive Involution.

KINETIC ARTS. The arts that evolve in time, music, poetry, etc., as contrasted with those that
exist in space (Static Arts). The latter, which are generally perceived by sight, are crystal-
lized in space and do not change in time. The kinetic arts, which do change in time after the
process of composition has been completed, are perceived by such sense organs as touch,
smell, taste, and, of course, hearing, Television and motion pictures are kinetic arts.

LOGARITHM. When a number is expressed as a power of ten, the exponent of that power is

called the logarithm to the base ten.
LOGARITHMIC SCALE. A series of points within two rational limits, selected by a process

of determining logarithmic ratios. Within the limits, b and a, intermediate points of uniform

I

K

symmetry may be designated as follows:

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681

m denotes determinant of a series.

MAJOR GENERATOR. The larger of two numbers or durations in the process of producing
Resultants by Interference.

MANIFOLD. Mathematically, a number of elements related under one system. In this system,
a Secondary Selective System, or one that is the result of selection and serves as the limit
of another selection. In music, e.g., a scale. In design, a color pattern, etc.

MECHANICAL PERMUTATION. See Displacement.

METHOD OF SERIES. A process of developing a related group of terms by inserting connecting
terms between given limits. Two terms become three, three become five, five become nine,
etc. Suppose we begin with Y and B. This becomes

Y G B

Y. . YG. G BG .B

A basic technique in Schillinger's approach.

MINOR GENERATOR. The smaller of two numbers or periodicities combined in the process
of Interference or Synchronization.

MONOMIAL. An expression consisting of a single term.

MONOMIAL PERIODICITY. A group or series composed of one number repeated several
times. Two monomial or uniform periodicities, in the process of Interference, produce a
non-uniform periodicity.

N

NATURAL HARMONIC SERIES. The series of overtones produced by a vibrating string or
air column. The original tone is known as the Fundamental Tone, and the overtones are
sometimes called Partials, because they are produced by vibrations of parts of the string.
When the C two octaves below middle C on the piano is struck, the fundamental tone is
C with a vibration frequency of 64. The first overtone or partial is C an octave higher,
with a frequency of 128. Next overtone is G, a fifth higher; then C, a fourth higher, with a
frequency of 256; then E, a major third higher; G, a major third higher; Bt>, a minor third
higher; C, major second; D, E, F#, G, At>, Bb, B; and the 16th in the series, C, five octaves
above the fundamental.

NATURAL SERIES. The series of natural integers; 1, 2, 3, 4, 5, 6, 7, 8, 9; and the natural frac-
tional series, Schillinger also refers to these two series as the Natural Har-
2 3 4 5 6 n

monic Series, which should not be confused with the Overtone series or Natural Harmonic
Series.

NOMOGRAPHY. The graph method of notation. In general, any scientific system for notat-
ing natural phenomena.

NORMAL SERIES. A series of numbers or terms evolved by the Method of Series.

O

OPERAND GROUP. Mathematical term for a melody, design pattern, color pattern, used as
the basis of an art work. In music, we have a tuning system (or Primary Selective System)
from which we select certain symmetric or asymmetric points known as scales (or Secondary
Selective System) from which in turn we abstract a melodic pattern (or operand group).

ORDINATE ROTATION. See Bl-Coordinate Rotation.

OVERTONE SERIES. See Natural Harmonic Series.

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i

P

$ The Greek letter Phi. Refers to process of phasic rotation. See Density.
p. r. Used to denote Positional Rotation.

PARAMETER. A measuring line. In the Cartesian sense, the bi-coordinate system of measur-
ing lines (horizontal and vertical) that make up a graph. In the Einsteim'an sense, a system
of measuring lines involving four coordinates: xi (width), xj (depth), xi (height), and x<
(time), Schillinger employs a special version of Einstein's correlated time-space parameters.
Time and space are regarded as general parameters, while special parameters are established
to measure each physical component of an art object appealing to a different organ of
sensation.

PARENT SCALE. A scale selected from our tuning system, from which a series of related scales

are derived through General and Circular Permutation.
PARTIALS. See Natural Harmonic Series.

PERIODICITY. The recurrence in time or space of some phenomenon — in time, of a note or
sound; in space, of an area, angle, arc, color, etc. The simplest periodicity is Monomial:
a + a + a + ... + a where each consecutive term is equivalent in extension (space) or
duration (time). In uniform periodicity, the repetitious factor may consist of one or more
terms.

PERIODIC MOTION. Commonly known as rhythm, periodic motion is regarded by Schillinger
as the basis of all art forms and of phenomena in the world of nature.

PERMUTATION. The process of modifying or varying the elements in a group by rearranging
their order or sequence. General permutation (also known as logical permutation) yields all
possible variations since it does not proceed in one direction as in the case of Circular Per-
mutation (or Displacement). A polynomial of 5 terms yields only 10 circular permutations
(5 clockwise and S counter-clockwise), but 120 general permutations (1X2X3X4X5). Both
figures include the original order as a permutation.

PERMUTATIONS OF A HIGHER ORDER. See Higher Order.

PHASIC ROTATION. A process for varying the density of a sound or space composition by
rotation or displacement along the time-axis, density-axis, or both: C G () Also
known as phasic displacement.
PITCH. The highness or lowness of a tone as determined by the dumber of vibrations per
second, or Frequency. In our tuning system, concert A has 440 vibrations per second. The
brilliant tone of the Boston Symphony is sometimes attributed to the fact that it tunes its
concert A at 444 vibrations per second.
PITCH-SCALE. A sequence of pitch-units selected from our tuning system according to some
definite pattern of increasing or decreasing frequencies. In Schillinger's terminology — a
Secondary Selective System as distinguished from our tuning system, which is a Primary-
Selective System. Schillinger does not follow the traditional system of classifying scales
as major and minor, but devises an exhaustive system of classification under four headings:
Group One: Scales with one tonic and consisting of any number of notes up to and in-
cluding a range of one octave.
Group Two; Scales with one tonic, a range of more than one octave, and evolved by Ei-

panding the scales in the first group.
Group Three: Scales of more than one tonic, a range of not more than one octave, and

constructed symmetrically.
Group Four: Scales of more than one tonic, a range of more than one octave, and con-
structed symmetrically.

POSITIONAL ROTATION. A general technique for varying the simultaneity phase (ordinate),
the continuity phase (abscissa), or both, of an art form. Involves the application of Circu-
lar Permutation to structures and sequences. In design, p. r. may produce superimposed
images. In music, p. r. produces variations in textural density. See Bi-coordinate Rotation
and Phasic Rotation,

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683

POLYNOMIAL. An algebraic expression containing two or more terms. In general, any group
made up of more than one element.

POWER SERIES. An acceleration series formed by raising the initial term to its successive
powers. 2, 4, 8, 16, 32 ... . is evolved by squaring 2, cubing 2, etc.

PRE-SET. The process of selecting the components of any art form in advance of the actual
composition, the specific components being chosen according to the effects desired.

PRIMARY SELECTIVE SYSTEM. In music our tuning system is the result of a selection
from the complete manifold (or dense set) of all possible frequencies. Schillinger designates
it a primary system. Mathematically speaking, such a system is a series of fixed points selected
symmetrically from the sound Continuum. A Secondary Selective System in turn involves
a selection from the primary system. In music, certain pitch-units are chosen to form a
scale. In design, certain colors or angles or arcs are selected to form color scales, etc.

PRIME NUMBER SERIES. An acceleration series composed of numbers that are divisible only
by 1 or themselves: 1, 2, 3, 5, 7, 11, 13, 17. .. .

PROGRESSIVE SYMMETRY. See Symmetry.

QUADRANT ROTATION. One of the fundamental techniques for varying an art form. The
original form denoted as (a), is developed backwards in time ©, then backwards in time and
upside down ©, and finally forward in time and upside down <3). The relation of these four
forms to each other is made clear by seeing them in relation to the four quadrants of a graph

and positive and negative fractions -, -, -, -, etc. Contrasts with Real Set, which

2 3 4 5

r Denotes Resultant.
Ra-rb* See a ~ b.

RATIONAL SET. Refers to numbers, specifically the positive and negative integers 0, 1,2, 3,

4, 5..

includes irrational numbers \/2, \/3, V5, \/2, as well as rational. A real set of numbers
refers to all the number values necessary to describe a Dense Set.

RATIONALIZATION. In design, the process of inscribing a structure, originally evolved in an
unbounded space, within a boundary. Also the process of subjecting a spatial form to the
tendency of its own ratio.

REAL SET. See Rational Set and Dense Set.

REGULARITY. The simplest form of regularity is uniformity or Monomial Periodicity. In
its more complex forms, regularity results from the combination of different uniformities.
Regularity is another word for Rhythm and the process of producing non-uniform forms of
regularity is Interference.

REGULARITY and COORDINATION. The foundation of Schillinger's approach to the arts
is his Theory of Regularity and Coordination, also known as the Theory of Rhythm in the
composition of music. This theory embraces the manipulation of duration, frequency, in-
tensity and quality factors underlying the process of artistic creation. The principle of rhythm
governs the periodicity (recurrence) of art components, from the most elementary, such as
attacks in music and extensions in design, to the most complex questions of form.

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GLOSSARY

RESULTANT. The product of the Interference of two or more uniform periodicities of differ-
ent frequencies, brought into synchronization. Such resultants may be obtained either by
graphs or direct computation, and are the parent-shapes of all rhythms. The interfering
periodicities are known as Generators.

RETARDATION SERIES. A series of numbers in which successive terms are the result of a
decreasing differential. See Acceleration Series.

RHYTHM. To Schillinger, rhythm is any form of periodic motion that may be discerned in
natural, social and artistic phenomena. Such periodic motion is reducible to a mathematically
conceivable Regularity, i.e., to numbers. Broadly speaking, rhythm is of two types: frac-
tional when it refers to units of a composition or design, and Factorial when it refers to the
structure of the whole.

RHYTHMIC CENTER. The center is generally regarded as a point that is equally distant from
the circumference of a circle or the sides of a rectangle. In contrast, the rhythmic center is
the result of ratios, not simple measurement.

RHYTHM ICON. See Appendix.

RUBATO. In the performance of music, an alteration in the duration of given notes, literally
by stealing time from other notes. Mathematically, rubato is accomplished by introducing a
standard unit of deviation and throwing a balanced binomial or polynomial out of balance.
4+4 thus becomes 5 + 3, and 2 + 2 + 2 + 2 becomes 1+3+2+2, etc.

S Used to denote Structural group in Positional Rotation, s denotes structural unit. S also
used to denote Synchronization, the coordination of durations, or Syrnmetrizatlon, the
coordination of extensions, as in S(a:b).

2 The Greek letter Sigma. Used to denote a large structure or something compounded of a group
of smaller structures. In music, the sigma is the same as the Expansion of a scale, save that
it is a chord, not a sequence. If we use numbers to denote the pitch-units (1, 2, 3, 4, 5, 6, 7, 8),
the 2 (or Ei) would be 1, J, 5, 7, 2(9), 4(11), 6(13). 2 is also used to denote a series.

SCALES OF LINEAR CONFIGURATION. In design, the equivalent of pitch-scales in music:
a sequence of linear forms selected from the time-space Continuum according to some
specific pattern of increasing or decreasing quantities, extensions, etc

SECONDARY SELECTIVE SYSTEM. Secondary systems vary in density and become iden-
tical with the primary system when they reach a saturation point. Full saturation of the
primary system in turn produces a Dense Set. In short, the creation of music or design in-
volves a series of selections. From the Continuum or total manifold, we select a series of
pitch-units which constitute our tuning system. From this primary selective system, we
make a second selection which yields a series of scales or secondary selective systems.
From these we make further selections, which result in melodies, color schemes, spatial
designs, etc.

SELECTIVE SYSTEM. See Primary Selective System and Secondary Selective System.

SEMANTICS. For Schillinger the study, not of the evolution of meaning, but of meaning itself.
More specifically, study of the relationship between form and sensation. Semantic require-
ments define the purpose or meaning of a work of art.

SERIES OF NATURAL DIFFERENCES. A series in which each succeeding term is based on
an increasing differential or difference: e.g., 1, 2, 4, 7, II, 16, 22, 29 The difference be-
tween 1 and 2 is 1 ; between 2 and 4 = 2; between 4 and 7 = 3; etc. Also known as an Accel-
eration Series.

SINE CURVE. The simple curve described by a pendulum swinging from a fixed point and leav-
ing a trace on a sheet of paper moving perpendicular to its line of motion. The curve, which
is regular and symmetrical, represents the motion of a simple sound wave, sometimes re-
ferred to as a sine wave. Sine curves differ considerably in appearance depending upon the

S

GLOSSARY

685

relation of Amplitude (height of a crest above the axis) and Wave-length (crest and trough).
A curve of the same general form but differing in phase by a quarter period, or 90°, is known
as a cosine curve.

AXIS

-COSINE CUBVE

SIIC CUM

SOLIDRAMA. See Appendix C.

SPLIT UNITS. The result of dividing a single unit in a Resultant or other harmonic series by
some divisor. The process of selecting the units to be split is itself controlled by Coefficients
of Recurrence or Permutation.

STATIC ARTS. See Kinetic Arts.

STYLE SERIES. A series of numbers that serves as the source of allrhvthmic patterns:-

n

Each of these yields a series of Determinants known

11112345

" "5* 4' 3'1'T' ? T* 1

2.3.4. n

as the - series, series, - series, - series. Rhythmic patterns derived from any one group

2 3 4 n

of determinants are known as a family, which accounts for Schillinger's occasional use of

5

Family-Series as an alternative for style series. If we take the - series, for example, we may

generate a full family of related Duration Groups. ^ may be split into 4 + 1 or 3 + 2. The

former binomial synchronized with itself produces the trinomial 1+3 + 1. This trinomial
synchronized with its various permutations produces 1+2 + 1+1 and 1+1+2 + 1.
All of these groups bear an apparent relation to each other and the parent series. Translated
into design or music components, such a series produces an identifiable style. Hybrid style
is based on several series, or several determinants, instead of one, which produces a pure
style.

SUMMATION SERIES. A series in which each number is the sum of the preceding two. From
an esthetic point of view, the most useful summation series are 1+2 = 3, 5, 8, 13, 21,

34, 57 1 + 3 = 4, 7, 11, 18, 29, 47, 76 Also known as Fibonacci Series after

Leonard of Pisa, who developed these additive series. Leonard was the son of Bonaccio
(Alius Bonacci), which gave the series its name.

SYMMETRIC SCALES. Pitch-Scales in music and Color Scales or Scales of Linear Con-
figurations in design, marked by uniformity of sequence and reversibility of pattern. Schil-
linger classifies pitch-scales into four groups, two of which are symmetrical. Group III, con-
fined to a range of less than one octave, contains, 2, 3, 4 and 6 tones between the units of
each scale. Group IV, with a range of more than one octave, contains 8, 9, 10 and 11 semi-
tones between units. Any of these scales is reversible: e.g., C — E — Ab — C, with 4 semi-
tones between each tone, contains these specific tones whether one begins at the bottom or
top.

SYMMETRY. The quality of a sequence, series, or form whereby the pattern is reversible.
The point around which the group balances or may be reversed is designated as the axis
of symmetry. In progressive symmetry the successive terms or elements, instead of being
identical in relation to a central point of reference, are marked by factors of growth: e.g.,
A + (A+B) + (A+B+C) + (B+C) + C.

Digiti

Google

Original from
UNIVERSITY OF MICHIGAN

686

GLOSSARY

SYNCHRONIZATION. The process of making two or more duration groups (or number series)
coincide as to period of lime. When the two series are not identical, Interference results.

T

T or t Symbol for time. Small t also used iq designate deviation unit. Capital T is also used to
denote a definite periodic portion of continuity along a line or extension, as T = ti -|-ta +
4"ti+ ' . t». In this formula, small t denotes duration units.

T a+ b. See a -S- b. Used interchangeably with R a -=-b.

r The Greek letter Tau. Denotes a unit of growth or a unit of deviation in the development

of asymmetrical patterns.
6 The Greek letter Theta. Used to denote a compound rotation group in the control of Density.

TEMPORAL SCALE. Any sequence of durations abstracted from the uniform set of natural
integers, Such a sequence may be developed through the process of Interference, by the
application of Coefficients of Recurrence, through Permutation, or Quadrant Rota-
tion.

TONIC. The root, starting point, or first tone of a Pitch-Scale. In Schillinger's classification of
scales there are systems of multiple tonics, e.g., two-tonic system (C and Fft, three-tonic
system (C, E, Ab), four-tonic system (C, Eb, Gb, A), six-tonic system (C, D, E, F#, G#,
A ft, and the twelve-tonic system (employing all the semitones in our equal temperament
tuning system).

TUNING SYSTEM. See Equal Temperament; also Primary Selective System.

U

UNIFORMITY. A concept associated with and evolved from various axiomatic proportions.
In the field of reasoning: the system of count based on the so-called natural integers. In the
field of sight: the system of measurement of space. And in the field of hearing: the clock
system of time measurement. Uniformity may be regarded as a special case of Periodicity.

V

VARIABLE VELOCITY, GROUPS OF. A general term for groups embodying various forms
of acceleration and growth, such as Arithmetical and Geometrical Progressions, Power
Series, Summation Series, etc,

W

WAVE-LENGTH. The length of the crest and trough of a wave produced by a vibrating medium
— sound, light or radio. While color in general depends upon wave-length, pitch is determined
by Frequency, or the number of waves per second. The oscillograph and phonodeick are
used to make photographic records of sound waves. >

i ■ * _ «

Diqilized by Google

Original from
UNIVERSITY OF MICHIGAN

INDEX

abscissa, 44, 234, 255

phases, 215

rotation, 255
abstract cinema, 68
acceleration, positive and negative, 44
accent, 41
accentuation, 24
Aeolian mode, 224
African drumming, 665
albumen, 254
algebraic powers, 173-183
American Indians, 98

American Institute for the Study of Advanced

Education, 46
American Institute of the City of New York,

Mathematics Division, 46
American Library of Musicology, 98
American Musicological Society, 46
American jazz, 140
angle units, 329-330
angle-perspective, 232
angles, symmetric construction of, 312
Angstrom units, 272
Annixter, Paul, 173
"antiquation," 253
Archipenko, 68
archipentura,

Archipenko, 62, 67, 68
architecture, Gothic, 57
arcs

alternating sin movement of, 320, 321
automatic continuity of, 341
moving in constant alternating direction,
317, 318

moving in constant direction, 314, 315

variable direction of, 336

variable lengths of, 324, 328
A Rebours (Huysmans), 80
Aristotle. 5, 13, 14, 27
Aristoxenes, 14
arithmetical progression, 636

series, 87, 88
art

ancient Egyptian, 89
ancient Greek, 89
creation and criteria of, 30-32
development of, 11
European, 43

experimental, 33

imitative, 35

kinetic, 67

static, 63
art and evolution, 7-10
art and nature, 3-6
art creation, intuitive period of, 5

pre-esthetic period of, 4, 5

rational period of, 5
art for art's sake, 17
art forms, 8, 10

complex, chart for combination of, 79

complex heterogeneous, 78

complex homogeneous, 74, 75

compound, 54

correspondences between, 80-84
diagram of, 60
first group, 58-71
individual table of, 59
kinetic, 74

measurable quantity of, 6
new, 663-667

technique of engineering in, 6
art, mathematics and, 38-47
art of audible sound, 64
art of audible word. 64
art of smellable odor, 65
art of tastable flavor, 66
art of texture of visible surface, 67
art of touchable mass, 65
art of visible light, 66
art of visible pigment, 66
art product, development of, by method of
normal series, 55

definition of. by method of series, 51-55
art production, the technology of, 3, 4
"artistic differential," 84
Artomaton, 674

arts, combined, production of, 429
arts, fine, 41
history of, 17

kinetic (musical), 61, 62, 63
liberal, 41

physical source of the, 18-22
static (plastic), 61, 62
technical, 41
Asia, music of, 221

attack group, synchronization of, 402, 403

687

rv -■ _j l nr\Ci\(> Original from

Digitized by VjUUgUw UNIVERSITY OF MICHIGAN

688

INDEX

auditory and visual forms, correlation of,

432-444
Aztec, 34

Bach, J. S., 16, 45, 55, 236, 237, 247, 400

balance, 184-192

barcarolles, 14

Barr, Professor, 10

beauty

mathematical definition of, 84

psychological definition of, 83
Beethoven, Ludwig van, 238, 239, 247. 400
Berg, 247
binomial, 95, 676

distributive cube of, 563-565

distributive square of, 502-519

elements, 168, 169

forms of standard deviation in, 153-156
groups, balanced, 140
groups, unbalanced, 140
growth of trinomials, 169, 170
. series, 96

Biological Bases of Evolution of Music, The

(Ivan Kryzhanovsky), 184
bithematic composition, 398
"blues," 26

B Minor Sonata (Liszt), 247
Brancusi, 70

calculus, differential, 365
Cartesian manner, 4
Caruso, Enrico, 20, 22
cathode-ray tube, 38
Catholic liturgical music, 186, 219
Caucasian rug patterns, 221
Cenozoic age, 8
Cezanne, 7

Chansons Madecasses (Ravel), 223
Charleston, 140, 141

music, 52
"Charles V, 1533" (painting), 233
Chinese pentatonic scales, 224
Chopin, 54, 247
chord structures, 12
chord progressions, 42

Chromatic Fantasy and Fugue (J. S. Bach),
247

cinema, 68, 215

cinematic design, time-space unit in, 429
"clavilux," 22
cochlea, 58
coefficient groups, 42
color
audition, 80

interpretation, 51
origin, 346

scales, 194, 274, 346-362

sectors, distribution of, 346

sequence, 215
color-axis, 346

angle of. 346
Columbia University, Faculty Club of (Math-
ematics Dept.), 669
combinatory analysis, 173
Compton, 365

Conditioned Reflexes (Ivan Pavlov), 58
configuration, 27, 39, 40, 45

scales, 44
consistency, 43
contemplative music, 14
continuity, 72, 85-108, 146, 215, 253, 260

composition of, 267

coordinate components, 85

definition of, 85

factorial-fractional, 92-98. 173

harmonic, 157

parametral components, 85

perfect form of, 265

rhythm of, 400

of rotary groups, 254
continuum, 51-84, 676

class, 85

definition of, 57

psycho-physiological, 146

space-time, 365

spatial, 39
contraction patterns, 28
contrapuntalists, 240
coordinate expansion, 44, 244-254
cortis arch, 58
Cortis organ, 27
counterpoint, 161, 215, 255

rhythmic, in graphic form, 161
cradle songs, 14
crescendo, 146
crystallization, 186

of event, 184-192
crystals, harmonic structure of, 4

Dada: Surrealism and Fantastic Art, exhibit,
33

Dali, 36
dance, 71

African, 73

Asiatic, 73

Balinese, 224
De Divina Proporzione (Luca Pacioli), 32
Debussy, 12, 45, 247

Google

Original from
UNIVERSITY OF MICHIGAN

INDEX

689

delta, 254, 255, 261

phasic rotation of, 262-266

variants of, 258
Democritus, 365
"dense set," 39, 40, 41, 43, 273
density, 45, 184

composition of, 254-270

in proportion to mobility, 266

scheme of, 255

variable, composition of from strata, 266-
270

density-groups, 45

binomial, 256, 257

compound, 254, 258, 261

monomial, 256

permutation of, 260

polynomial, 257

simultaneous, 254

sequent, 254

smaller than A, 259
density-time relations, 256
density-unit, 254, 256, 266
derivative series, 113
Des Esseintes, 80
Descartes, 365
design analysis, 375-390
design,

geometrical variation of, 373

ornamental, African, 43

ornamental, Asiatic, 43

linoleum, 389, 390

problems in, 397, ,398

production of, 363-398

rug, 387

stereometric, 74

wall paper, 388
design, linear, 73, 74

elements of, 367

variation of, 369
design, rhythmic, 365-374

definition of, 365

evolution of. through elements of, 372
determinant, 98-107

series with one, 101
deviation

in binomial or polynomial, 153, 154, 156

series of, 149-153
diffused polygon, 32
"diminished 7th" chord, 247
Dirac, Professor, 7
discontinuity, 218
discontinuous 180° arcs, 313
displacement, 158-162

graphic representation of, 159, 160

displacement, 163

arranged in simultaneity, 159, 160
distributive

cubing, 415

involution, 43, 173, 416-418
involution groups, 502-635
involution in linear design, 418-421
properties, 173
squaring, 414
"Djanger," 224

Double Equal Temperament, 273, 274, 665
Duerer, 7

duration group, 402-404, 678

distribution of, 400-402

synchronized, distribution of, 404-405
duration-units, 255

Dynamic Symmetry (Jay Hambidge), 32
dynamic symmetry, theory of, 89

ecclesiastic music, 14
Eddington, 365
Eiffel Tower, 31
Einstein, Albert, 56, 73, 365
Einsteinian manner, 4
El Greco, 45, 245
electro-magnetic behaviour, 56
electro-mechanical synthesis, 47
Empire State Building, 30, 250
equal temperament system, 273, 277
equilibrium, unstable, 184-192
esthetic experience

varieties of, 14, 15, 16
esthetic expression, the semantics of, 3, 4
esthetic satisfaction, 84
esthetic symbols, nature of, 25
Euclidean geometry, 73
evolutionary series, method of interference,
43

expansion

coordinate, 244-254

geometrical, of harmonized melody, 253

group, 248

patterns, 28

pitch, 248-251

tonal, 248
expansion-contraction series, 148
exponent scale, 272

factorial continuity,

determination of, 93
factorial- fractional continuity, 679
Fall of the House of Usher (Watson and

Webber), 69
family series, 679

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

690 IN

Fibonacci (Leonard of Pisa), 33
"Fibonacci series," 88, 679
folklore, 98
formalism, 14

Forms of Primitive Music (Helen Roberts),
98

forms of unity
logical association, 38
perceptive auditory association, 38
perceptive visual association, 38

fractional series, natural, plotted, 87

fractional continuity, determination of. 93

Franciscan Order, 219

frequency, 19, 41, 147

fresco, 67

Fugues (Bach), SS
futurists, 67

gamelan, 3

General and Special Theory of Relativity

(Albert Einstein), 56
general determinant,

cubing of, 135
generator

in music, 52

in photography, 52
geometrical expansions, 244, 251, 252
geometrical inversion, 232, 236, 238, 240
geometrical progression, 636

series, 88
geometry, 345

fluent, 57
Gershwin, George, 238
Gibbon, 173
Golden Gate Bridge, 30
Goldschmidt, 4
graph, musical, 244
graph, unit of measurement of, 244
"graphomaton," 674
Greeks, ancient, 52

music of, 223

Hambidge, Jay, 32, 89
Handel, 45, 247
harmonic accompaniment, 255
harmonic contrasts, 43

coordination and continuity of, 173
harmonic motion, simple, 109

graphed, 1 10
harmonic ratios, 442
harmonic relations, 34, 35, 146

and harmonic coordination, 41^4
harmonic series, natural, 247
harmonic waves, 365

harmonic whole, 35
harmonics, 24, 680
harpsichord, 55
Haydn, Joseph, 238, 250
Hebrews, 186
Hellenic, 43

Helmholtz, Hermann Ludwig Ferdinand von,

24, 57
Herzog, George, 25
Hindemith, 45, 247
Hinton, 74

History oj Music (Karl Stork), 25
honeycomb, 67, 221

cells, 43
Hungary, folk dances of, 146
Huysmans, 80
hymns, revolutionary, 14

Indians, Oklahoma, music of, 220

Institute of Light, Grand Central Palace,

New York, 69
instrumental group, synchronization of, 406,

407

instrumental interference, composition of re-
sultant of, 408, 409
integer series, fractional, 86-87

natural, 85, 86

natural, plotted, 86
intensity, 19, 42
interference, 365, 447

of periodicities, 366
interior decorating, 67
intonation, 25

International Exposition in Paris (1937), 16
inversions, geometric, 234, 235, 253
inverted symmetry, 140
involution

distributive, 274

groups, 43

series, 44
Isolde's Love-Death (Wagner), 28
Ivan the Terrible Killing His Son (Repin).
33

Kandinsky, 33, 36
Kant, 58
Kasner, 4

kinetic art of light projected on 3>dimen-
sional or 2-dimensional screen in motion.
70

kinetic art of visible mass, 7 1
kinetic art of visible pigment, 71
kinetic art of visible pigment transforming
on moving surface, 68, 69

Cnr\n\(> Original from

^ uu d' UNIVERSITY OF MICHIGAN

INDEX

691

kinetic art of visible texture of surface or

volume, 71
kinetic arts, 680

kinetic design, production of, 414-427
kinetic geometry, 50, 345
kinetic light of visible art projected on plane
surface, 68

kinetic linear design, elementary illustration
of. 442

kinetic sequence of image, 215
kinetic visual forms, 36
Klee, 33
Klein. 68
Krenek, 247

Kryzhanovsky, Ivan, 184

Leibnitz, 365
Leonardo da Vinci, 32

"Les Fontaines Lumineuses," Paris Interna-
tional Exposition, 1937, 70
Liebling, Leonard, 238
li-ki, 6

linear configurations, scales of saturation, 385,
386

linear design, 42
compositions in, 392-396
definition of, 363

distributive involution in, 418-422

elements of, 363, 364

illustration of proportions, 245

rhythmic groups in, 335, 336
Lipchitz, Jacques, collection (Paris), 233
Liszt, Franz, 52, 247
Lobachevsky, 73
logarithm, 680

logarithmetic contraction of time, 232

dependence of ratios, 39, 273

selection, 272
Lorentz, 365
love songs, 14
lumia, 16, 22
"luminaton," 674

Mach, 40

Madagascar, 223

magic, 11, 37

major genetic factor, 447

Marinetti; 61

master patterns, 44, 45

Mathematicians Faculty Club of Columbia

University, 46
Mathematics Museum of Teachers College,

Columbia University, 46
melodic curve, 23

melodic trajectory, 23
melody, 255

analysis of, 23

definition of, 24, 29

dualism of problem of, 23

function of, 25

geometrical expansion of, 251

insufficiency of, 25

semantics of, 25-28
memorial rites of the ancient Chinese, 6
Mendel, 43
metronome, 274

Metropolitan Museum of Art, 246

Michael Angelo, 32

Michelson, 365

military music, 52

Miller, D. C, 64

Millikan, 365

mimicry, 11

Minkowsky, 56, 365

minor genetic factor, 447

mobility, density in proportion to, 266

"modernization," 253

Modigliani, 245

"Mogen Dovid," Hebrew, 186, 219
monodic musical culture, 399
monomial, 94

Moonlight Sonata (L. van Beethoven), 247

Moore, Prof. Douglas, 28

mosaics, mobile, 68

motion picture, 36

Mozart, 400

Musamaton, 673

Museum of Modem Art, 33, 246
music

African, 43

Arabian, 26

Asiatic, 43

analysis of dramatic qualities of, 57
Chinese, 25, 41
elements of, 432
European, 140, 219
expression, 232
Greek, ancient, 223
medicinal application of, 14
modern school, 247
natural sources of, 13
of civilized world, averages in, 97
of civilized world, series of averages in,
98

production of,, 399-413
real meaning of, 26, 27
reflexological origin of, 26
musical instruments, development of, 11

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

692

INDEX

natural fraction and integer series plotted to-
gether, 90
natural harmonic series, 636
natural integers, 38, 43
natural integer and fraction series plotted to-
gether, 90
natural series

fractional continuity, 92

fractional continuity, rational ratios of. 91
Navajo sand paintings, 36
Neanderthal man, 8
New School for Social Research, 46
Newton, 365

New York University, 46
New York World's Fair, 16
Niagara Falls, 67, 70
1922 (Piano Suite)
(Hindemith), 247
nocturnes, 14
notation, 232
Nova lis. 15

operand group, 272

optical illusion, 73

optical perspective, 146

optical projection, through extension of ordi-
nate, 244

orchestral writing, 255

ordinate, 255
phases, 215
rotation, 255, 681

organ, electronic, 665

organic art, 32
nature of, 32-34

Origin of Music, The (Karl Stumpf), 13

oscillations, 52, 109
transverse, of a pendulum (illus.), 109

oscillogram, 23, 56

oscillograph, 38

overtones, 146

Facioli, Luca, 32
paint, luminescent, 66
"painted sculpture," 69
painting, 67

Navajo sand, 36
parameter, 56, 57

correlated, 4

frequency, 64

general, 146

intensity, 64

quality, 64

special, 146
parametral interpretation of a system, 56-58

Paris International Exposition, 70

"partials," 24, 682

pastorals, 14

Pavlov, Ivan, 27, 58

"pelog," 224

pendulum, 109

pentacle, 10

pentagonal form, sequence of angles in, 220

perception, 39
esthetic and selective,' 30

periodic motion, 37

periodic regularity, 41

periodic series
binomial, binomial relations of, 129-134
binomial, polynomial relations of, 136, 137
polynomial, binomial relations of, 137, 138
polynomial, polynomial relations of, 129-
140

practical application in art, 140-141

synchronization of, 129
periodic waves, 41
periodicities

monomial, polynomial relations of, 123-
129

monomial, synchronization of three, 124
monomial, synchronization of several, 123,
125

monomial, synchronization of two, 112, 113
simultaneous monomial, 112-123
periodicity, 27, 109-157, 288, 682
consecutive displacement of, 120
derivative, 114, 120, 121
factorial, 174
fractional, 174, 175
in general parameters, 146
in phases, 109
in special parameters, 146
monomial, 110

monomial, relation of two (illus,), 111
monomial, of sector radii, 339
of angles, 293-313
of angles, binomial, 297
of angles, monomial, 293
of angles, trinomial, 301, 312
of arcs and radii, 324-328
of dimensions, 284-292
of expansion and contraction, 146-156
of radii and angle values, 314-323
of rectilinear segments, binomial, 287, 288
of rectilinear segments, trinomial, 2SS
synchronization and derivative, 115
permutation, 42, 141, 158-172
binomial, 168
circular, 158, 215

Google

Original from
UNIVERSITY OF MICHIGAN

INDEX

693

general or logical, 162
in mechanical sequence, tables of, 164-168
mechanical, 162, 163
of a higher order, 168
of four elements, mechanical scheme for,
172

of sequent density-groups, 260-261
Persia, 221

Persian popular songs, 224
phase, 109

arrangement, 340
phase-units, 40
phasic differences, 18
phasic rotation, 682

of delta, 262-266
phonograms, 665
photography, 67
Phrygian scale, 240

Physical Optics (Robert W. Wood), 80
"Piano Sonata No. 8" (Ludwig van Beet-
hoven), 238, 239
Picasso, Pablo, 33
Pissarro, 32
pitch, 24, 682
axis, 252

contraction of, 247, 248

discrimination, 54

expansion of, 244-248

frequencies, 40

geometrical expansion of, 45

musical, 51
pitch-coordinate, 45
pitch-scales, 42, 682

Arabian, 223
pitch-unit, 244, 247, 255

scale of, and corresponding expansions,
253
planets, 223
planimetric clusters, 69
planimetric linear design, 73
Plato, 14, 27, 52
"poetic image," 29

"Poetry and Prose Mathematically Devised,"
669

poetic structure, elements of, 670
poetry, structural unit of, 670
"pointillism," 35, 67
Politeia (Plato), 14, 52
polygons, closed, conceived as monomial pe-
riodicity of angles, dimensions and direc-
tions, 344-345
polynomial

distributive cube of, 586

distributive property of, 173

distributive square of, 542
forms of standard deviation in, 153
groups, balanced, 140
series, 96

Porgy and Bess (George Gershwin), 238
positional rotation, 45, 215-218, 423-428, 682

definition of, 215

dimensionality of, 215-218

in design, 215

in music, 215
positive and negative values, series, 97
power series, 637

powers, general treatment of, 179-183
primary selective series, 42
primary selective systems, 38, 39, 40
prime number, 10

series, 90, 637
progressions

arithmetic, 44, 636

geometric, 44, 636
progressive series, 44
Prokofieff, 252

proportionate distribution within rectangular

areas, 414-418
Ptolemy, 365
"pyramids," 400
Pythagoras, 15

Puerto Rico, dance, songs of, 141

quadrant rotation, 44, 232-243, 272
quintinomials

cube of, 589-613

squares of, 543-561

range-contraction, 252, 253
Raphael, 52

ratio and rationalization, 193-214
ratio of radii and scale of curvature of arcs,
328

ratio, uniform, series, 277

ratio-realization of space, 193

rational behaviour, 193

rational continuum, ratios of, 107-108

rational composition, 193

rational set, 272

rational values, factorial-fractional continuity

of, 97
Ravel, 12, 223
real set, 272

rectangle, rationalization of, 195-214
rectilinear segments, 302, 304, 305

infinite series, 288-290

symmetric contraction of, 312
reflexological origin of music, 26

Digitized by GoOgle

Original from
UNIVERSITY OF MICHIGAN

694

INDEX

regularity

and coordination, theory of, 34, 41, 49
law of, 109

pragmatic validity of theory of, 45-46
Repin, 33

Republic (Plato), 27
resultant, 44
revolutionary songs, 14
rhythm, 37, 39, 41, 147, 400, 684

attacks of, 42

coordinators of, 400

definition of, 109

intensities of, 42

of durations, 42, 274

of variable velocities, 636, 637

qualities of, 42

renaissance of, 400

sequences of chord progressions of, 42
spatial, 431
temporal, 431
the laws of, 4
theory of, 265
rhythmic
center, 336, 337, 33ft, 370,371
composition of sentences, 670
durations, 141

groups in linear design, 329-336
patterns, 11, 12

reality, expansion-contraction as a, 148
resultants, 275

resultants, with fractioning, 27S-276

series of deviation, 149
Rhythmicon, 274, 665

-produced forms, 666, 667
Riemann, 56, 73, 365
Rimington, 68
Roberts, Helen, 98
Rome, 221
root-tone, 252

rotary phases, sequence of, 262
rotation

bi- coordinate, 255

of phases, 255

scales of, 262
rotation-phase, compound, 254
rubato, 153

Russia, European, music of, 221
Russian folk music, 140, 223

"St. Anthony of Padua" (painting), 233
Saint- Martin, 15
Sappho, 7

scale of twelve hues, 351-362
scale units, 40

scales, 44
Aeolian, 223
assymetric, 278
Balinese, 224
color, 274, 346-362
derivative, 42
Dorian, 223
expanded, 278, 280
Hindu, 39

Javanese, 39, 223, 224

Locrian, 223

Lydian, 223

Mixolydian, 223

of linear configuration, 274, 684

of linear configuration and area, 284-345

Phrygian, 223

pitch, 274, 277, 278, 279, 280, 2SI, 282
sonic, 670

symmetric, 278, 281, 282, 283

symmetrical, 278

temporal, 274
Sckillinger System oj Musical Composition,
The (Joseph Schillinger), 23, 232, 244,
254, 277. 674
Schlegel, Friedrich, 57
scholasticism, 14
Schopenhauer, 15
science and esthetics, 1-47
Science of Musical Sounds, The (D, C. Mil-
ler), 64
screen (illustration), 429
screen, distributive squaring of, 430. 431
sculpture, 70
Eea urchin, 220
Seashore, Prof. Carl, 58
secondary selective series, 42
secondary selective systems, 46, 684
sector radii, 313
selective systems, 273-362, 684

primary, 273, 274

secondary, 273, 274
"selenders," 224
semantics, 23, 25, 26
semitone, 244

Sensations of Tone (Hermann von Helm-

holtz), 57
sensory perceptions, 58
sequence of simultaneity, 161
serenades, 14
serial determinant, 43
series of natural differences, 89-90
series of prime numbers, 90
Seurat, Georges, 32, 35
Shakespeare, 7

Google

Original from
UNIVERSITY OF MICHIGAN

INDEX

695

Shaw, Arnold, vii, 675
shiva, 34
sight, sense of, 68
2 sigma, 267, 268
simple harmonic motion, 365
simultaneity, 215

rhythm of, 400
sine curve, 110, 684
sine waves, 18, 24, 27, 109
"slendro," 224
sliding pitch, 146
"slow motion," 33, 146
snowflake formations, 222
solidrama, 71

forms of, 667
Solidrive, 71, 667
sound, 164

animal, 13

biological factors of, 13

cinema, 54

description of, 58

frequency, 54

waves, 58, 109, 146
sounding texture, 254
sources of art of music, 5-6
space theory, 50

"space-time continuum," 39, 40, 41
space-time relations, 58
spatial design, 146
spatial kinetic configuration, 43
spectrum, 51, 146

full, 346, 351

full (illus.), 347
Spencer, 15

Sposalizio (Franz Liszt), 52
Sposalizio (Raphael), 52
star (illustration), 221
starfish, 43, 220
static art

of 3-dimensional visible mass, 70
of visible light placed inside 3-dimensional

spatial form, 69
of visible pigment covering surfaces of 3-

dimensional form, 69
of visible texture of 3-dimensional forms,
69

static optical forms, 63
Steiner, Ralph, 69
Stony Indians

expressions in song, 25
Stork, Karl, 25
strata, 255

composition of variable density from, 266-
270

structure

density of, 184

in stable equilibrium, 184
Stumpf, Karl, 13
style determinants, 141
summation series, 34, 44, 88, 89, 637
superimposition, 267
supersonic waves, 37
Surf and Seaweed (Ralph Steiner), 69
surrealism, 7, 33
symmetrization, 447
symmetric parallelisms, 219-222
symmetric scales, 685
symmetry, 34, 219-231

bifold, 223

dynamic, 32, 34

esthetic evaluation on the basis of, 223-
224

progressive, 157

rectangular (extensions), 224-231
synchronization, 112, 447, 686
' binary, 274, 447-457
binary and ternary, 445-501
binary, with fractioning, 458-475
double process of, 145
of a motive, 134
of a musical motive, 135
of different rhythmic series, 144-146
of rhythmic series with itself, 141-143
of second order, 141-146
ternary, 274, 476

Tactilism (Marinetti), 61
tangent trajectories, 4

Teachers College of Columbia University
(Depts. of Mathematics, Fine Arts and
Music), 46
telecasting, 47
television, 16, 47
temporal plasticity, 397
temporal structures, coordination of, 399-401
thematic

components, 146

entity, 140

motif, 42

textures, 255
Theory of Regularity, 42
Theory of Relativity, 56, 73
Theremin, Leon, 58, 665
theta, 254
time

as general parameter, 71-74

positions implied by conception of, 232

psycho-physical, 72

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696

INDEX

Titania Palast, Berlin, 69
tonic, 686
trajectory, 73, 74
transformer, 52
transmitter, 53
triads, 241
trinomials

distributive cube of, 568-576

distributive square of, 520-541
tuning, 277

Javanese, 221
tuning fork, 24, 38
tuning system, 273, 274

contraction of, 45
Two-Part Inventions (J. S. Bach), 236,

ultra-violet rays, 66
uniformity, 38, 39

and primary selective systems, 38-41
unstable equilibrium, 254

variable velocity groups of, 274, 636-641
variation and composition, techniques, 44-45

velocities, 44

constant, 147
vertical coincidence, 340
Vinci, Leonardo da, 16
visual and auditory forms, correlation of,

432-444
visual art, 45, 232

visual kinetic composition, elements of, 432

Wagner, 28
Watson, 69

wave-length, color and sound, 21
wave motion, 37
Webber, 69
237 Webem, Anton von, 247, ,250, 2S2
Whitlock, 4

Wilfred, Thomas, 22, 68
With the Greatest of Ease (Paul Annixter),
173

Wood, Robert W,, 80
Woolworth, 65

"zer ef kend," 219, 220, 223

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